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Altaic languages | Uralo-Altaic hypothesis | Uralo-Altaic hypothesis
In 1844, the Finnish philologist Matthias Castrén proposed a broader grouping which later came to be called the Ural–Altaic family, which included Turkic, Mongolian, and Manchu-Tungus (=Tungusic) as an "Altaic" branch, and also the Finno-Ugric and Samoyedic languages as the "Uralic" branch (though Castrén himself used the terms "Tataric" and "Chudic"). The name "Altaic" referred to the Altai Mountains in East-Central Asia, which are approximately the center of the geographic range of the three main families. The name "Uralic" referred to the Ural Mountains.
While the Ural-Altaic family hypothesis can still be found in some encyclopedias, atlases, and similar general references, since the 1960s it has been heavily criticized. Even linguists who accept the basic Altaic family, such as Sergei Starostin, completely discard the inclusion of the "Uralic" branch.
The term continues to be used for the central Eurasian typological, grammatical and lexical convergence zone.BROWN, Keith and OGILVIE, Sarah eds.:Concise Encyclopedia of Languages of the World. 2009. p. 722. Indeed, "Ural-Altaic" may be preferable to "Altaic" in this sense. For example, Juha Janhunen states that "speaking of 'Altaic' instead of 'Ural-Altaic' is a misconception, for there are no areal or typological features that are specific to 'Altaic' without Uralic." |
Altaic languages | Korean and Japanese languages | Korean and Japanese languages
In 1857, the Austrian scholar Anton Boller suggested adding Japanese to the Ural–Altaic family.Roy Andrew Miller (1986): Nihongo: In Defence of Japanese. .
In the 1920s, G.J. Ramstedt and E.D. Polivanov advocated the inclusion of Korean. Decades later, in his 1952 book, Ramstedt rejected the Ural–Altaic hypothesis but again included Korean in Altaic, an inclusion followed by most leading Altaicists (supporters of the theory) to date.Gustaf John Ramstedt (1952): Einführung in die altaische Sprachwissenschaft ("Introduction to Altaic Linguistics"). Volume I, Lautlehre ("Phonology"). His book contained the first comprehensive attempt to identify regular correspondences among the sound systems within the Altaic language families.
In 1960, Nicholas Poppe published what was in effect a heavily revised version of Ramstedt's volume on phonologyNicholas Poppe (1960): Vergleichende Grammatik der altaischen Sprachen. Teil I. Vergleichende Lautlehre, ('Comparative Grammar of the Altaic Languages, Part 1: Comparative Phonology'). Wiesbaden: Otto Harrassowitz. (Only part to appear of a projected larger work.)Roy Andrew Miller (1991): "Genetic connections among the Altaic languages." In Sydney M. Lamb and E. Douglas Mitchell (editors), Sprung from Some Common Source: Investigations into the Prehistory of Languages, 1991, 293–327. . that has since set the standard in Altaic studies. Poppe considered the issue of the relationship of Korean to Turkic-Mongolic-Tungusic not settled. In his view, there were three possibilities: (1) Korean did not belong with the other three genealogically, but had been influenced by an Altaic substratum; (2) Korean was related to the other three at the same level they were related to each other; (3) Korean had split off from the other three before they underwent a series of characteristic changes.
Roy Andrew Miller's 1971 book Japanese and the Other Altaic Languages convinced most Altaicists that Japanese also belonged to Altaic.Nicholas Poppe (1976): "Review of Karl H. Menges, Altajische Studien II. Japanisch und Altajisch (1975)". In The Journal of Japanese Studies, volume 2, issue 2, pages 470–474.Roy Andrew Miller (1971): Japanese and the Other Altaic Languages. University of Chicago Press. . Since then, the "Macro-Altaic" has been generally assumed to include Turkic, Mongolic, Tungusic, Korean, and Japanese.
In 1990, Unger, emphasizing the need to establish language relationships rigorously "from the bottom up," advocated comparing Tungusic with the common ancestor of Korean and Japanese before seeking connections with Turkic or Mongolic.J. Marshall Unger (1990): "Summary report of the Altaic panel." In Philip Baldi, ed., Linguistic Change and Reconstruction Methodology, pages 479–482. Mouton de Gruyter, Berlin.
However, many linguists dispute the alleged affinities of Korean and Japanese to the other three groups. Some authors instead tried to connect Japanese to the Austronesian languages.
In 2017, Martine Robbeets proposed that Japanese (and possibly Korean) originated as a hybrid language. She proposed that the ancestral home of the Turkic, Mongolic, and Tungusic languages was somewhere in northwestern Manchuria. A group of those proto-Altaic ("Transeurasian") speakers would have migrated south into the modern Liaoning province, where they would have been mostly assimilated by an agricultural community with an Austronesian-like language. The fusion of the two languages would have resulted in proto-Japanese and proto-Korean.Martine Irma Robbeets (2017): "Austronesian influence and Transeurasian ancestry in Japanese: A case of farming/language dispersal". Language Dynamics and Change, volume 7, issue 2, pages 201–251, Martine Irma Robbeets (2015): Diachrony of verb morphology – Japanese and the Transeurasian languages. Mouton de Gruyter.
In a typological study that does not directly evaluate the validity of the Altaic hypothesis, Yurayong and Szeto (2020) discuss for Koreanic and Japonic the stages of convergence to the Altaic typological model and subsequent divergence from that model, which resulted in the present typological similarity between Koreanic and Japonic. They state that both are "still so different from the Core Altaic languages that we can even speak of an independent Japanese-Korean type of grammar. Given also that there is neither a strong proof of common Proto-Altaic lexical items nor solid regular sound correspondences but, rather, only lexical and structural borrowings between languages of the Altaic typology, our results indirectly speak in favour of a “Paleo-Asiatic” origin of the Japonic and Koreanic languages." |
Altaic languages | The Ainu language | The Ainu language
In 1962, John C. Street proposed an alternative classification, with Turkic-Mongolic-Tungusic in one grouping and Korean-Japanese-Ainu in another, joined in what he designated as the "North Asiatic" family.John C. Street (1962): "Review of N. Poppe, Vergleichende Grammatik der altaischen Sprachen, Teil I (1960)". Language, volume 38, pages 92–98. The inclusion of Ainu was adopted also by James Patrie in 1982.James Tyrone Patrie (1978): The genetic relationship of the Ainu language. PhD thesis, University of Hawaii.James Tyrone Patrie (1982): The Genetic Relationship of the Ainu Language. University of Hawaii Press.
The Turkic-Mongolic-Tungusic and Korean-Japanese-Ainu groupings were also posited in 2000–2002 by Joseph Greenberg. However, he treated them as independent members of a larger family, which he termed Eurasiatic.Joseph Greenberg (2000–2002): Indo-European and Its Closest Relatives: The Eurasiatic Language Family, 2 volumes. Stanford University Press.
The inclusion of Ainu is not widely accepted by Altaicists. In fact, no convincing genealogical relationship between Ainu and any other language family has been demonstrated, and it is generally regarded as a language isolate.
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Altaic languages | Early criticism and rejection | Early criticism and rejection
Starting in the late 1950s, some linguists became increasingly critical of even the minimal Altaic family hypothesis, disputing the alleged evidence of genetic connection between Turkic, Mongolic and Tungusic languages.
Among the earlier critics were Gerard Clauson (1956), Gerhard Doerfer (1963), and Alexander Shcherbak. They claimed that the words and features shared by Turkic, Mongolic, and Tungusic languages were for the most part borrowings and that the rest could be attributed to chance resemblances. In 1988, Doerfer again rejected all the genetic claims over these major groups.Gerhard Doerfer (1988): Grundwort und Sprachmischung: Eine Untersuchung an Hand von Körperteilbezeichnungen. Franz Steiner. Wiesbaden: |
Altaic languages | Modern controversy | Modern controversy
A major continuing supporter of the Altaic hypothesis has been Sergei Starostin, who published a comparative lexical analysis of the Altaic languages in 1991. He concluded that the analysis supported the Altaic grouping, although it was "older than most other language families in Eurasia, such as Indo-European or Finno-Ugric, and this is the reason why the modern Altaic languages preserve few common elements".
In 1991 and again in 1996, Roy Miller defended the Altaic hypothesis and claimed that the criticisms of Clauson and Doerfer apply exclusively to the lexical correspondences, whereas the most pressing evidence for the theory is the similarities in verbal morphology.Roy Andrew Miller (1991), page 298Roy Andrew Miller (1996): Languages and History: Japanese, Korean and Altaic. Oslo: Institute for Comparative Research in Human Culture. . Pages 98–99
In 2003, Claus Schönig published a critical overview of the history of the Altaic hypothesis up to that time, siding with the earlier criticisms of Clauson, Doerfer, and Shcherbak.
In 2003, Starostin, Anna Dybo and Oleg Mudrak published the Etymological Dictionary of the Altaic Languages, which expanded the 1991 lexical lists and added other phonological and grammatical arguments.
Starostin's book was criticized by Stefan Georg in 2004 and 2005,Stefan Georg (2004): "[Review of Etymological Dictionary of the Altaic Languages (2003)]". Diachronica volume 21, issue 2, pages 445–450. Stefan Georg (2005): "Reply (to Starostin response, 2005)". Diachronica volume 22, issue 2, pages 455–457. and by Alexander Vovin in 2005.Alexander Vovin (2005): "The end of the Altaic controversy" [review of Starostin et al. (2003)]. Central Asiatic Journal volume 49, issue 1, pages 71–132.
Other defenses of the theory, in response to the criticisms of Georg and Vovin, were published by Starostin in 2005,Sergei A. Starostin (2005): "Response to Stefan Georg's review of the Etymological Dictionary of the Altaic Languages". Diachronica volume 22, issue 2, pages 451–454. Blažek in 2006,Václav Blažek (2006): "Current progress in Altaic etymology." Linguistica Online, 30 January 2006. Accessed on 2019-03-22. Robbeets in 2007,Martine Robbeets (2007): "How the actional suffix chain connects Japanese to Altaic." In Turkic Languages, volume 11, issue 1, pages 3–58. and Dybo and G. Starostin in 2008.Anna V. Dybo and Georgiy S. Starostin (2008): "In defense of the comparative method, or the end of the Vovin controversy." Aspects of Comparative Linguistics, volume 3, pages 109–258. RSUH Publishers, Moscow
In 2010, Lars Johanson echoed Miller's 1996 rebuttal to the critics, and called for a muting of the polemic.Lars Johanson (2010): "The high and low spirits of Transeurasian language studies" in Johanson and Robbeets, eds. Transeurasian Verbal Morphology in a Comparative Perspective: Genealogy, Contact, Chance., pages 7–20. Harrassowitz, Wiesbaden. Quote: "The dark age of pro and contra slogans, unfair polemics, and humiliations is not yet completely over and done with, but there seems to be some hope for a more constructive discussion." |
Altaic languages | List of supporters and critics of the Altaic hypothesis | List of supporters and critics of the Altaic hypothesis
The list below comprises linguists who have worked specifically on the Altaic problem since the publication of the first volume of Ramstedt's Einführung in 1952. The dates given are those of works concerning Altaic. For supporters of the theory, the version of Altaic they favor is given at the end of the entry, if other than the prevailing one of Turkic–Mongolic–Tungusic–Korean–Japanese. |
Altaic languages | Major supporters | Major supporters
Pentti Aalto (1955). Turkic–Mongolic–Tungusic–Korean.
Anna V. Dybo (S. Starostin et al. 2003, A. Dybo and G. Starostin 2008).
Frederik Kortlandt (2010).
Karl H. Menges (1975). Common ancestor of Korean, Japanese and traditional Altaic dated back to the 7th or 8th millennium BC (1975: 125).
Roy Andrew Miller (1971, 1980, 1986, 1996). Supported the inclusion of Korean and Japanese.
Oleg A. Mudrak (S. Starostin et al. 2003).
Nicholas Poppe (1965). Turkic–Mongolic–Tungusic and perhaps Korean.
Alexis Manaster Ramer.
Peter Benjamin Golden
Martine Robbeets (2004, 2005, 2007, 2008, 2015, 2021) (in the form of "Transeurasian").
G. J. Ramstedt (1952–1957). Turkic–Mongolic–Tungusic–Korean.
George Starostin (A. Dybo and G. Starostin 2008).
Sergei Starostin (1991, S. Starostin et al. 2003).
John C. Street (1962). Turkic–Mongolic–Tungusic and Korean–Japanese–Ainu, grouped as "North Asiatic".
Talât Tekin (1994). Turkic–Mongolic–Tungusic–Korean. |
Altaic languages | Major critics | Major critics
Gerard Clauson (1956, 1959, 1962)
Gerhard Doerfer (1963, 1966, 1967, 1968, 1972, 1973, 1974, 1975, 1981, 1985, 1988, 1993)
Susumu Ōno (1970, 2000)
Juha Janhunen (1992, 1995) (tentative support of Mongolic-Tungusic)
Claus Schönig (2003)
Stefan Georg (2004, 2005)
Alexander Vovin (2005, 2010, 2017) - Formerly an advocate of Altaic (1994, 1995, 1997, 1999, 2000, 2001), later a critic
Alexander Shcherbak
Alexander B. M. Stiven (2008, 2010) |
Altaic languages | Advocates of alternative hypotheses | Advocates of alternative hypotheses
James Patrie (1982) and Joseph Greenberg (2000–2002). Turkic–Mongolic–Tungusic and Korean–Japanese–Ainu, grouped in a common taxon (cf. John C. Street 1962).
J. Marshall Unger (1990). Tungusic–Korean–Japanese ("Macro-Tungusic"), with Turkic and Mongolic as separate language families.
Lars Johanson (2010). Agnostic, proponent of a "Transeurasian" verbal morphology not necessarily genealogically linked. |
Altaic languages | "Transeurasian" renaming | "Transeurasian" renaming
In Robbeets and Johanson (2010), there was a proposal to replace the name "Altaic" with the name "Transeurasian". While "Altaic" has sometimes included Japonic, Koreanic, and other languages or families, but only on the consideration of particular authors, "Transeurasian" was specifically intended to always include Turkic, Mongolic, Tungusic, Japonic, and Koreanic. Robbeets and Johanson gave as their reasoning for the new term: 1) to avoid confusion between the different uses of Altaic as to which group of languages is included, 2) to reduce the counterproductive polarization between "Pro-Altaists" and "Anti-Altaists"; 3) to broaden the applicability of the term because the suffix -ic implies affinity while -an leaves room for an areal hypothesis; and 4) to eliminate the reference to the Altai mountains as a potential homeland.Martin Robbeets & Alexander Savelyev. "Introduction," The Oxford Guide to the Transeurasian Languages (2020, Oxford University Press), page 1.
In Robbeets and Savelyev, ed. (2020) there was a concerted effort to distinguish "Altaic" as a subgroup of "Transeurasian" consisting only of Turkic, Mongolic, and Tungusic, while retaining "Transeurasian" as "Altaic" plus Japonic and Koreanic. |
Altaic languages | Arguments | Arguments |
Altaic languages | For the Altaic grouping | For the Altaic grouping |
Altaic languages | Phonological and grammatical features | Phonological and grammatical features
The original arguments for grouping the "micro-Altaic" languages within a Uralo-Altaic family were based on such shared features as vowel harmony and agglutination.
According to Roy Miller, the most pressing evidence for the theory is the similarities in verbal morphology.
The Etymological Dictionary by Starostin and others (2003) proposes a set of sound change laws that would explain the evolution from Proto-Altaic to the descendant languages. For example, although most of today's Altaic languages have vowel harmony, Proto-Altaic as reconstructed by them lacked it; instead, various vowel assimilations between the first and second syllables of words occurred in Turkic, Mongolic, Tungusic, Korean, and Japonic. They also included a number of grammatical correspondences between the languages. |
Altaic languages | Shared lexicon | Shared lexicon
Starostin claimed in 1991 that the members of the proposed Altaic group shared about 15–20% of apparent cognates within a 110-word Swadesh-Yakhontov list; in particular, Turkic–Mongolic 20%, Turkic–Tungusic 18%, Turkic–Korean 17%, Mongolic–Tungusic 22%, Mongolic–Korean 16%, and Tungusic–Korean 21%.Sergei A. Starostin (1991): Altajskaja problema i proisxoždenie japonskogo jazyka ('The Altaic Problem and the Origin of the Japanese Language'). Nauka, Moscow. The 2003 Etymological Dictionary includes a list of 2,800 proposed cognate sets, as well as a few important changes to the reconstruction of Proto-Altaic. The authors tried hard to distinguish loans between Turkic and Mongolic and between Mongolic and Tungusic from cognates; and suggest words that occur in Turkic and Tungusic but not in Mongolic. All other combinations between the five branches also occur in the book. It lists 144 items of shared basic vocabulary, including words for such items as 'eye', 'ear', 'neck', 'bone', 'blood', 'water', 'stone', 'sun', and 'two'.Sergei Starostin, Anna V. Dybo, and Oleg A. Mudrak (2003): Etymological Dictionary of the Altaic Languages, 3 volumes. .
Robbeets and Bouckaert (2018) use Bayesian phylolinguistic methods to argue for the coherence of the "narrow" Altaic languages (Turkic, Mongolic, and Tungusic) together with Japonic and Koreanic, which they refer to as the Transeurasian languages.Robbeets, M.; Bouckaert, R.: Bayesian phylolinguistics reveals the internal structure of the Transeurasian family. Journal of Language Evolution 3 (2), pp. 145–162 (2018) Their results include the following phylogenetic tree:Structure of Transeurasian language family revealed by computational linguistic methods . 2018. Max Planck Institute for the Science of Human History.
Martine Robbeets et al. (2021) argues that early Transeurasian speakers were originally agriculturalists in Northeastern Asia, only becoming pastoralists later on.
The analysis conducted by Kassian et al. (2021) on a 110-item word list, specifically developed for each of the languages — Proto-Turkic, Proto-Mongolic, Proto-Tungusic, Middle Korean and Proto-Japonic — indicated partial support for the Altaic macrofamily, with Korean seemingly excluded. While acknowledging that prehistoric contacts are a plausible alternative explanation for the positive results, they deem such a scenario less likely for the lexical matches between Turkic and Japonic languages, which are better explained by genealogical relationship because of the substantial geographical distances involved. Quote: "Korean shows no positive results with any of its potential Altaic relatives....Korean emerges as either unrelated to any of these four taxa or impervious to the efficacy of the algorithm owing to major mutations undergone by non-initial consonants in Pre-Proto-Korean." |
Altaic languages | Against the grouping | Against the grouping |
Altaic languages | Weakness of lexical and typological data | Weakness of lexical and typological data
According to G. Clauson (1956), G. Doerfer (1963), and A. Shcherbak (1963), many of the typological features of the supposed Altaic languages, particularly agglutinative strongly suffixing morphology and subject–object–verb (SOV) word order,Hawkins and Gilligan (1988): "The suffixing preference", in The Final-Over-Final Condition: A Syntactic Universal, page 326. MIT Press. ; According to the table, among the surveyed languages, 75% of OV languages are mainly suffixing, and more than 70% of mainly suffixing languages are OV. often occur together in languages.
Those critics also argued that the words and features shared by Turkic, Mongolic, and Tungusic languages were for the most part borrowings and that the rest could be attributed to chance resemblances. They noted that there was little vocabulary shared by Turkic and Tungusic languages, though more shared with Mongolic languages. They reasoned that, if all three families had a common ancestor, we should expect losses to happen at random, and not only at the geographical margins of the family; and that the observed pattern is consistent with borrowing.
According to C. Schönig (2003), after accounting for areal effects, the shared lexicon that could have a common genetic origin was reduced to a small number of monosyllabic lexical roots, including the personal pronouns and a few other deictic and auxiliary items, whose sharing could be explained in other ways; not the kind of sharing expected in cases of genetic relationship.Schönig (2003): "Turko-Mongolic Relations." In The Mongolic Languages, edited by Juha Janhunen, pages 403–419. Routledge. |
Altaic languages | The Sprachbund hypothesis | The Sprachbund hypothesis
Instead of a common genetic origin, Clauson, Doerfer, and Shcherbak proposed (in 1956–1966) that Turkic, Mongolic, and Tungusic languages form a Sprachbund: a set of languages with similarities due to convergence through intensive borrowing and long contact, rather than common origin.Gerard Clauson (1956). "The case against the Altaic theory". Central Asiatic Journal volume 2, pages 181–187Gerhard Doerfer (1963): "Bemerkungen zur Verwandtschaft der sog. altaische Sprachen" ('Remarks on the relationship of the so-called Altaic languages') In Gerhard Doerfer ed.: Türkische und mongolische Elemente im Neupersischen, Bd. I: Mongolische Elemente im Neupersischen, pages 51–105. Franz Steiner, WiesbadenAlexander Shcherbak (1963).
Asya Pereltsvaig further observed in 2011 that, in general, genetically related languages and families tend to diverge over time: the earlier forms are more similar than modern forms. However, she claims that an analysis of the earliest written records of Mongolic and Turkic languages shows the opposite, suggesting that they do not share a common traceable ancestor, but rather have become more similar through language contact and areal effects. |
Altaic languages | Hypothesis about the original homeland<span class="anchor" id="Postulated Urheimat"></span> | Hypothesis about the original homeland
The prehistory of the peoples speaking the "Altaic" languages is largely unknown. Whereas for certain other language families, such as the speakers of Indo-European, Uralic, and Austronesian, it is possible to frame substantial hypotheses, in the case of the proposed Altaic family much remains to be done.Miller (1991), page 319–320
Some scholars have hypothesised a possible Uralic and Altaic homeland in the Central Asian steppes.Nikoloz Silagadze, "The Homeland Problem of Indo-European Language-Speaking Peoples", 2010. Faculty of Humanities at Ivane Javakhishvili Tbilisi State University. .Y.N. Matyuishin (2003), pages 368–372.
thumb|right|Hypothesized homeland according to Blench (2009)
Chaubey and van Driem propose that the dispersal of ancient Altaic language communities is reflected by the early Holocene dissemination of haplogroup C2 (M217): "If the paternal lineage C2 (M217) is correlated with Altaic linguistic affinity, as appears to be the case for Turkic, Mongolic and Tungusic, then Japanese is no Father Tongue, and neither is Korean. This Y-chromosomal haplogroup accounts for 11% of Korean paternal lineages, and the frequency of the lineage is even more reduced in Japan. Yet this molecular marker may still be a tracer for the introduction of Altaic language to the archipelago, where the paternal lineage has persisted, albeit in a frequency of just 6%."Gyaneshwer Chaubey and George van Driem (2020) Munda languages are father tongues, but Japanese and Korean are not. (p. 11)
thumb|362x362px|Detailed tree of the Altaic languages.
Juha Janhunen hypothesized that the ancestral languages of Turkic, Mongolic, Tungusic, Korean, and Japanese were spoken in a relatively small area comprising present-day North Korea, Southern Manchuria, and Southeastern Mongolia.Lars Johanson and Martine Irma Robbeets (2010): Transeurasian Verbal Morphology in a Comparative Perspective: Genealogy, Contact, Chance.. Introduction to the book, pages 1–5. However Janhunen is sceptical about an affiliation of Japanese to Altaic,Juha Janhunen (1992): "Das Japanische in vergleichender Sicht". Journal de la Société finno-ougrienne, volume 84, pages 145–161. while András Róna-Tas remarked that a relationship between Altaic and Japanese, if it ever existed, must be more remote than the relationship of any two of the Indo-European languages.András Róna-Tas (1988). Ramsey stated that "the genetic relationship between Korean and Japanese, if it in fact exists, is probably more complex and distant than we can imagine on the basis of our present state of knowledge".S. Robert Ramsey (2004): "Accent, Liquids, and the Search for a Common Origin for Korean and Japanese". Japanese Language and Literature, volume 38, issue 2, page 340. American Association of Teachers of Japanese.
Supporters of the Altaic hypothesis formerly set the date of the Proto-Altaic language at around 4000 BC, but today at around 5000 BC or 6000 BC.Elena E. Kuz'mina (2007): The Origin of the Indo-Iranians, page 364. Brill. This would make Altaic a language family older than Indo-European (around 3000 to 4000 BC according to mainstream hypotheses) but considerably younger than Afroasiatic (c. 10,000 BCIgor M. Diakonoff (1988): Afrasian Languages. Nauka, Moscow. or 11,000 to 16,000 BCEhret (2002) according to different sources). |
Altaic languages | See also | See also
Classification of the Japonic languages
Nostratic languages
Pan-Turanism
Turco-Mongol
Uralo-Siberian languages
Xiongnu
Comparison of Japanese and Korean |
Altaic languages | References | References |
Altaic languages | Citations | Citations |
Altaic languages | Sources | Sources
Aalto, Pentti. 1955. "On the Altaic initial *p-." Central Asiatic Journal 1, 9–16.
Anonymous. 2008. [title missing]. Bulletin of the Society for the Study of the Indigenous Languages of the Americas, 31 March 2008, 264: ____.
Anthony, David W. 2007. The Horse, the Wheel, and Language. Princeton: Princeton University Press.
Boller, Anton. 1857. Nachweis, daß das Japanische zum ural-altaischen Stamme gehört. Wien.
Clauson, Gerard. 1959. "The case for the Altaic theory examined." Akten des vierundzwanzigsten internationalen Orientalisten-Kongresses, edited by H. Franke. Wiesbaden: Deutsche Morgenländische Gesellschaft, in Komission bei Franz Steiner Verlag.
Clauson, Gerard. 1968. "A lexicostatistical appraisal of the Altaic theory." Central Asiatic Journal 13: 1–23.
Doerfer, Gerhard. 1973. "Lautgesetze und Zufall: Betrachtungen zum Omnicomparativismus." Innsbrucker Beiträge zur Sprachwissenschaft 10.
Doerfer, Gerhard. 1974. "Ist das Japanische mit den altaischen Sprachen verwandt?" Zeitschrift der Deutschen Morgenländischen Gesellschaft 114.1.
Doerfer, Gerhard. 1985. Mongolica-Tungusica. Wiesbaden: Otto Harrassowitz.
Georg, Stefan. 1999 / 2000. "Haupt und Glieder der altaischen Hypothese: die Körperteilbezeichnungen im Türkischen, Mongolischen und Tungusischen" ('Head and members of the Altaic hypothesis: The body-part designations in Turkic, Mongolic, and Tungusic'). Ural-altaische Jahrbücher, neue Folge B 16, 143–182.
.
Lee, Ki-Moon and S. Robert Ramsey. 2011. A History of the Korean Language. Cambridge: Cambridge University Press.
Menges, Karl. H. 1975. Altajische Studien II. Japanisch und Altajisch. Wiesbaden: Franz Steiner Verlag.
Miller, Roy Andrew. 1980. Origins of the Japanese Language: Lectures in Japan during the Academic Year 1977–1978. Seattle: University of Washington Press. .
Ramstedt, G.J. 1952. Einführung in die altaische Sprachwissenschaft I. Lautlehre, 'Introduction to Altaic Linguistics, Volume 1: Phonology', edited and published by Pentti Aalto. Helsinki: Suomalais-Ugrilainen Seura.
Ramstedt, G.J. 1957. Einführung in die altaische Sprachwissenschaft II. Formenlehre, 'Introduction to Altaic Linguistics, Volume 2: Morphology', edited and published by Pentti Aalto. Helsinki: Suomalais-Ugrilainen Seura.
Ramstedt, G.J. 1966. Einführung in die altaische Sprachwissenschaft III. Register, 'Introduction to Altaic Linguistics, Volume 3: Index', edited and published by Pentti Aalto. Helsinki: Suomalais-Ugrilainen Seura.
Robbeets, Martine. 2004. "Swadesh 100 on Japanese, Korean and Altaic." Tokyo University Linguistic Papers, TULIP 23, 99–118.
Robbeets, Martine. 2005. Is Japanese related to Korean, Tungusic, Mongolic and Turkic? Wiesbaden: Otto Harrassowitz.
Strahlenberg, P.J.T. von. 1730. Das nord- und ostliche Theil von Europa und Asia.... Stockholm. (Reprint: 1975. Studia Uralo-Altaica. Szeged and Amsterdam.)
Strahlenberg, P.J.T. von. 1738. Russia, Siberia and Great Tartary, an Historico-geographical Description of the North and Eastern Parts of Europe and Asia.... (Reprint: 1970. New York: Arno Press.) English translation of the previous.
Tekin, Talat. 1994. "Altaic languages." In The Encyclopedia of Language and Linguistics, Vol. 1, edited by R.E. Asher. Oxford and New York: Pergamon Press.
Vovin, Alexander. 1993. "About the phonetic value of the Middle Korean grapheme ᅀ." Bulletin of the School of Oriental and African Studies 56(2), 247–259.
Vovin, Alexander. 1994. "Genetic affiliation of Japanese and methodology of linguistic comparison." Journal de la Société finno-ougrienne 85, 241–256.
Vovin, Alexander. 2001. "Japanese, Korean, and Tungusic: evidence for genetic relationship from verbal morphology." Altaic Affinities (Proceedings of the 40th Meeting of PIAC, Provo, Utah, 1997), edited by David B. Honey and David C. Wright, 83–202. Indiana University, Research Institute for Inner Asian Studies.
Vovin, Alexander. 2010. Koreo-Japonica: A Re-Evaluation of a Common Genetic Origin. University of Hawaii Press.
Whitney Coolidge, Jennifer. 2005. Southern Turkmenistan in the Neolithic: A Petrographic Case Study. Oxbow Books. |
Altaic languages | Further reading | Further reading
Blažek, Václav. "Altaic numerals". In: Blažek, Václav. Numerals: comparative-etymological analyses of numeral systems and their implications: (Saharan, Nubian, Egyptian, Berber, Kartvelian, Uralic, Altaic and Indo-European languages). Vyd. 1. V Brně: Masarykova univerzita, 1999, pp. 102–140. ;
Dybo, Anna. "New trends in European studies on the Altaic problem". In: Journal of Language Relationship 14, no. 1-2 (2017): 71–106. https://doi.org/10.31826/jlr-2017-141-208
Finch, Roger. "Gender Distinctions in Nouns and Pronouns of the Altaic Languages". Expressions of Gender in the Altaic World: Proceedings of the 56th Annual Meeting of the Permanent International Altaistic Conference (PIAC), Kocaeli, Turkey, July 7–12, 2013. Edited by Münevver Tekcan and Oliver Corff. Berlin, Boston: De Gruyter, 2021. pp. 57–84. https://doi.org/10.1515/9783110748789-008
Greenberg, Joseph H. 1997. "Does Altaic exist?". In: Irén Hegedus, Peter A. Michalove, and Alexis Manaster Ramer (editors), Indo-European, Nostratic and Beyond: A Festschrift for Vitaly V. Shevoroshkin, Washington, DC: Institute for the Study of Man, 1997, 88–93. (Reprinted in Joseph H. Greenberg, Genetic Linguistics, Oxford: Oxford University Press, 2005, 325–330.)
Hahn, Reinhard F. 1994. LINGUIST List 5.908, 18 August 1994.
Janhunen, Juha. 1995. "Prolegomena to a Comparative Analysis of Mongolic and Tungusic". Proceedings of the 38th Permanent International Altaistic Conference (PIAC), 209–218. Wiesbaden: Harrassowitz.
Janhunen, Juha A. 2023. "The Unity and Diversity of Altaic", Annual Review of Linguistics 9:135–154 (January 2023)
Johanson, Lars. 1999. "Cognates and copies in Altaic verb derivation". In: Language and Literature – Japanese and the Other Altaic Languages: Studies in Honour of Roy Andrew Miller on His 75th Birthday, edited by Karl H. Menges and Nelly Naumann, 1–13. Wiesbaden: Otto Harrassowitz. (Also: HTML version.)
Johanson, Lars. 1999. "Attractiveness and relatedness: Notes on Turkic language contacts". Proceedings of the Twenty-Fifth Annual Meeting of the Berkeley Linguistics Society: Special Session on Caucasian, Dravidian, and Turkic Linguistics, edited by Jeff Good and Alan C.L. Yu, 87–94. Berkeley: Berkeley Linguistics Society.
Johanson, Lars. 2002. Structural Factors in Turkic Language Contacts, translated by Vanessa Karam. Richmond, Surrey: Curzon Press.
Kortlandt, Frederik. 1993. "The origin of the Japanese and Korean accent systems". Acta Linguistica Hafniensia 26, 57–65.
Robbeets, Martine. 2004. "Belief or argument? The classification of the Japanese language." Eurasia Newsletter 8. Graduate School of Letters, Kyoto University.
Ruhlen, Merritt. 1987. A Guide to the World's Languages. Stanford University Press.
Sinor, Denis. 1990. Essays in Comparative Altaic Linguistics. Bloomington: Indiana University, Research Institute for Inner Asian Studies. .
Vovin, Alexander. 2009. "Japanese, Korean, and other 'non-Altaic' languages". In: Central Asiatic Journal 53 (1): 105–147.
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Altaic languages | External links | External links
Swadesh vocabulary lists for Altaic languages (from Wiktionary's Swadesh-list appendix)
Monumenta altaica Altaic linguistics website, maintained by Ilya Gruntov
Altaic Etymological Dictionary, database version by Sergei A. Starostin, Anna V. Dybo, and Oleg A. Mudrak (does not include introductory chapters)
LINGUIST List 5.911 defense of Altaic by Alexis Manaster Ramer (1994)
LINGUIST List 5.926 1. Remarks by Alexander Vovin. 2. Clarification by J. Marshall Unger. (1994)
Category:Agglutinative languages
Category:Central Asia
Category:Eurocentrism
Category:Proposed language families
Category:Sprachbund |
Altaic languages | Table of Content | Short description, Earliest attestations<span class="anchor" id="Earliest attestations of the languages"></span>, History of the Altaic family concept, Origins, Uralo-Altaic hypothesis, Korean and Japanese languages, The Ainu language, Early criticism and rejection, Modern controversy, List of supporters and critics of the Altaic hypothesis, Major supporters, Major critics, Advocates of alternative hypotheses, "Transeurasian" renaming, Arguments, For the Altaic grouping, Phonological and grammatical features, Shared lexicon, Against the grouping, Weakness of lexical and typological data, The Sprachbund hypothesis, Hypothesis about the original homeland<span class="anchor" id="Postulated Urheimat"></span>, See also, References, Citations, Sources, Further reading, External links |
Austrian German | short description | Austrian German (), Austrian Standard German (ASG), Standard Austrian German (), Austrian High German (), or simply just Austrian (), is the variety of Standard German written and spoken in Austria and South Tyrol.Dollinger, Stefan. 2021. Österreichisches Deutsch oder Deutsch in Österreich? Identitäten im 21. Jahrhundert. 3rd ed. Vienna: nap, p. 14, https://www.nid-library.com/Home/ViewBook/512/16/view It has the highest sociolinguistic prestige locally, as it is the variation used in the media and for other formal situations. In less formal situations, Austrians use Bavarian and Alemannic dialects, which are traditionally spoken but rarely written in Austria. It has been standardized with the publishing of the Österreichisches Wörterbuch in 1951. |
Austrian German | History | History
Austrian German has its beginning in the mid-18th century, when Empress Maria Theresa and her son Joseph II introduced compulsory schooling in 1774, and several reforms of administration in their multilingual Habsburg Empire. At the time, the written standard was Oberdeutsche Schreibsprache (Upper German written language), which was highly influenced by the Bavarian and Alemannic dialects of Austria. Another option was to create a new standard based on the Southern German dialects, as proposed by the linguist Johann Siegmund Popowitsch. Instead they decided for pragmatic reasons to adopt the already-standardized chancellery language of Saxony (Sächsische Kanzleisprache or Meißner Kanzleideutsch), which was based on the administrative language of the non-Austrian area of Meißen and Dresden.
Austria High German (Hochdeutsch in Österreich, not to be confused with the Bavarian Austria German dialects) has the same geographic origin as the Swiss High German (Schweizer Hochdeutsch, not to be confused with the Alemannic Swiss German dialects).
The process of introducing the new written standard was led by Joseph von Sonnenfels.
Since 1951, the standardized form of Austrian German for official governmental use and in schools has been defined by the ("Austrian Dictionary"), published originally at the behest of the Austrian Federal Ministry of Education, Arts and Culture (in the 1950s the "Unterrichtsministerium", under minister Felix Hurdes) with Verlag Jugend & Volk, then by the Österreichischer Bundesverlag. |
Austrian German | Standard Austrian German | Standard Austrian German
The German language is a plurientric language and Austrian German is one of its standardized forms. The official Austrian dictionary, , prescribes spelling rules that define the official language.
Austrian delegates participated in the international working group that drafted the German spelling reform of 1996 and several conferences leading up to the reform were hosted in Vienna at the invitation of the Austrian federal government. Austria adopted it as a signatory, along with Germany, Switzerland, and Liechtenstein, of an international memorandum of understanding () signed in Vienna in 1996.
The eszett (ß) is used in Austria and Germany but not in Switzerland. In Austria, it is usually only called "scharfes s" ("sharp s").
right|thumb| (1995), an Austrian primary-school handwriting style
thumb|A sign in Vienna: ("pedestrian") is in Germany. In all-caps words, capital ẞ (instead of SS) became standard in both nations in 2017, but SS remains valid.
Distinctions in vocabulary persist, for example, in culinary terms, for which communication with Germans is frequently difficult, and administrative and legal language because of Austria's exclusion from the development of a German nation-state in the late 19th century and its manifold particular traditions. A comprehensive collection of Austrian-German legal, administrative and economic terms is offered in Markhardt, Heidemarie: Wörterbuch der österreichischen Rechts-, Wirtschafts- und Verwaltungsterminologie (Peter Lang, 2006).
Because of German's pluricentric nature, German dialects in Austria should not be confused with the variety of Standard Austrian German spoken by most Austrians, which is distinct from that of Germany or Switzerland. In the field of German dialectology, the notion of Standard Austrian German has been both debated and defended by German linguists since the 1970s. A One Standard German Axiom, effectively preventing the development of newer standards of German, has recently been offered as a characteristic of the field but remains to be discussed discipline-internally.Dollinger, S. (2024). Eberhard Kranzmayer’s dovetailing with Nazism: His fascist years and the ‘One Standard German Axiom (OSGA)’. Discourse & Society, 36(2), 147-179. https://doi.org/10.1177/09579265241259094 (Original work published 2025) |
Austrian German | Former spoken standard | Former spoken standard
Until 1918, the spoken standard in Austria was the , a sociolect spoken by the imperial Habsburg family and the nobility of Austria-Hungary. The sociolect, a variety of Standard German, is influenced by Viennese German and other Austro-Bavarian dialects spoken in eastern Austria but is slightly nasalized. |
Austrian German | Special written forms | Special written forms
For many years, Austria had a special form of the language for official government documents that is known as , or "Austrian chancellery language". It is a very traditional form of the language, probably derived from medieval deeds and documents, and has a very complex structure and vocabulary generally reserved for such documents. For most speakers (even native speakers), this form of the language is generally difficult to understand, as it contains many highly specialised terms for diplomatic, internal, official, and military matters. There are no regional variations because the special written form has been used mainly by a government that has now for centuries been based in Vienna.
is now used less and less because of various administrative reforms that reduced the number of traditional civil servants (). As a result, Standard Austrian German is replacing it in government and administrative texts. |
Austrian German | European Union | European Union
When Austria became a member of the European Union on 1 January 1995, 23 food-related terms were listed in its accession agreement as having the same legal status as the equivalent terms used in Germany,
for example, the words for "potato", "tomato", and "Brussels sprouts". (Examples in "Vocabulary")
Austrian German is the only variety of a pluricentric language recognized under international law or EU primary law.Markhardt's Das österreichische Deutsch im Rahmen der EU, Peter Lang, 2005. The focus on food-related vocabulary in "Protocol 23" is owed to trade requirements and therefore utterly accidental.De Cillia, Rudolf. 1998. "Burenwurst bleibt Burenwurst": Sprachpolitik und Gesellschaftliche Mehrsprachigkeit in Österreich. Klagenfurt: Drava. |
Austrian German | Grammar | Grammar |
Austrian German | Verbs | Verbs
In Austria, as in the German-speaking parts of Switzerland and in southern Germany, verbs that express a state tend to use as the auxiliary verb in the perfect, as well as verbs of movement. Verbs which fall into this category include (to sit), (to lie) and, in parts of Styria and Carinthia, (to sleep). Therefore, the perfect of these verbs would be , and , respectively.
In Germany, the words (to stand) and (to confess) are identical in the present perfect: . The Austrian variant avoids that potential ambiguity ( from , "to stand"; and from , "to confess": ).
In addition, the preterite (simple past) is very rarely used in Austria, especially in the spoken language, with the exception of some modal verbs (, ). |
Austrian German | Vocabulary | Vocabulary
There are many official terms that differ in Austrian German from their usage in most parts of Germany. Words used in Austria are (January) rather than , (more rare than Jänner) in variation with , (this year) along with , (stairs) along with , (chimney) instead of , many administrative, legal and political terms, and many food terms, including the following:Otto Back, Erich Benedikt, Karl Blüml, et al.: Österreichisches Wörterbuch (neue Rechtschreibung). Herausgegeben im Auftrag des Bundesministeriums für Unterricht, Kunst und Kultur. Auf der Grundlage des amtlichen Regelwerks. 41. circulation, Österreichischer Bundesverlag, Wien 2009,
Austrian Standard GermanStandard GermanEnglishBrandteigkrapferlWindbeutelCream puffEierspeise Rühreier Scrambled eggsErdapfel (also Bavarian and Southern German) Kartoffel PotatoFaschiertes Hackfleisch Minced meat/Ground beefFisolen Gartenbohnen or Grüne Bohnen Common beans /green beansKarfiol (also Bavarian and Southern German) Blumenkohl CauliflowerKohlsprossen Rosenkohl Brussel sproutsKren (also Bavarian and Southern German) Meerrettich HorseradishKukuruz (southeastern and western Austria)Mais Maize/cornMarille Aprikose ApricotMelangeMilchkaffeeMilk heavy coffee drinkMelanzaniAubergineAubergine/eggplantPalatschinke Pfannkuchen PancakeParadeiser (Vienna, Eastern Austria)Tomate TomatoPfefferoniPeperoni or ChiliChili pepperRote Rübe Rote Bete BeetrootSauce Tartare Remoulade Tartar SauceSchlagobers Schlagsahne Whipped creamStanitzelEiswaffelIce cream coneStaubzuckerPuderzuckerIcing sugar/powdered sugarTopfen (also Bavarian) Quark Quark, a semi-sweet cottage cheeseWeckerl (also Bavarian)BrötchenRoll (bread)
There are, however, some false friends between the two regional varieties:
(wardrobe) along with or instead of (and, similarly, along with , fridge), as opposed to (box) instead of . in Germany means both "box" and "chest".
(chair) instead of . means "" in Germany and means "stool (faeces)" in both varieties. |
Austrian German | Dialects | Dialects |
Austrian German | Classification | Classification
Dialects of the Austro-Bavarian group, which also comprises dialects from Bavaria
Central Austro-Bavarian (along the main rivers Isar and Danube, spoken in the northern parts of the State of Salzburg, Upper Austria, Lower Austria, and northern Burgenland)
Viennese German
Southern Austro-Bavarian (in Tyrol, South Tyrol, Carinthia, Styria, and the southern parts of Salzburg and Burgenland)
Vorarlbergerisch, spoken in Vorarlberg, is a High Alemannic dialect. |
Austrian German | Regional accents | Regional accents
In addition to the standard variety, in everyday life most Austrians speak one of a number of Upper German dialects.
While strong forms of the various dialects are not fully mutually intelligible to northern Germans, communication is much easier in Bavaria, especially rural areas, where the Bavarian dialect still predominates as the mother tongue. The Central Austro-Bavarian dialects are more intelligible to speakers of Standard German than the Southern Austro-Bavarian dialects of Tyrol.
Viennese, the Austro-Bavarian dialect of Vienna, is seen for many in Germany as quintessentially Austrian. The people of Graz, the capital of Styria, speak yet another dialect which is not very Styrian and more easily understood by people from other parts of Austria than other Styrian dialects, for example from western Styria.
Simple words in the various dialects are very similar, but pronunciation is distinct for each and, after listening to a few spoken words, it may be possible for an Austrian to realise which dialect is being spoken. However, in regard to the dialects of the deeper valleys of the Tyrol, other Tyroleans are often unable to understand them. Speakers from the different provinces of Austria can easily be distinguished from each other by their particular accents (probably more so than Bavarians), those of Carinthia, Styria, Vienna, Upper Austria, and the Tyrol being very characteristic. Speakers from those regions, even those speaking Standard German, can usually be easily identified by their accent, even by an untrained listener.
Several of the dialects have been influenced by contact with non-Germanic linguistic groups, such as the dialect of Carinthia, where, in the past, many speakers were bilingual (and, in the southeastern portions of the state, many still are even today) with Slovene, and the dialect of Vienna, which has been influenced by immigration during the Austro-Hungarian period, particularly from what is today the Czech Republic. The German dialects of South Tyrol have been influenced by local Romance languages, particularly noticeable with the many loanwords from Italian and Ladin.
The geographic borderlines between the different accents (isoglosses) coincide strongly with the borders of the states and also with the border with Bavaria, with Bavarians having a markedly different rhythm of speech in spite of the linguistic similarities. |
Austrian German | References | References |
Austrian German | Notes | Notes |
Austrian German | Citations | Citations |
Austrian German | Works cited | Works cited |
Austrian German | Further reading | Further reading
Ammon, Ulrich: Die deutsche Sprache in Deutschland, Österreich und der Schweiz: Das Problem der nationalen Varietäten. de Gruyter, Berlin/New York 1995.
Ammon, Ulrich / Hans Bickel, Jakob Ebner u. a.: Variantenwörterbuch des Deutschen. Die Standardsprache in Österreich, der Schweiz und Deutschland sowie in Liechtenstein, Luxemburg, Ostbelgien und Südtirol. Berlin/New York 2004, .
Dollinger, Stefan: Österreichisches Deutsch oder Deutsch in Österreich? Identitäten im 21. Jahrhundert. New Academic Press, 2021. Available online, 3rd ed.:https://www.nid-library.com/Home/BookDetail/512
Grzega, Joachim: „Deutschländisch und Österreichisches Deutsch: Mehr Unterschiede als nur in Wortschatz und Aussprache.“ In: Joachim Grzega: Sprachwissenschaft ohne Fachchinesisch. Shaker, Aachen 2001, S. 7–26. .
Grzega, Joachim: "On the Description of National Varieties: Examples from (German and Austrian) German and (English and American) English". In: Linguistik Online 7 (2000).
Grzega, Joachim: "Nonchalance als Merkmal des Österreichischen Deutsch". In: Muttersprache 113 (2003): 242–254.
Muhr, Rudolf / Schrodt, Richard: Österreichisches Deutsch und andere nationale Varietäten plurizentrischer Sprachen in Europa. Wien, 1997
Muhr, Rudolf/Schrodt, Richard/Wiesinger, Peter (eds.): Österreichisches Deutsch: Linguistische, sozialpsychologische und sprachpolitische Aspekte einer nationalen Variante des Deutschen. Wien, 1995.
Pohl, Heinz Dieter: „Österreichische Identität und österreichisches Deutsch“ aus dem „Kärntner Jahrbuch für Politik 1999“
Wiesinger, Peter: Die deutsche Sprache in Österreich. Eine Einführung, In: Wiesinger (Hg.): Das österreichische Deutsch. Schriften zur deutschen Sprache. Band 12. (Wien, Köln, Graz, 1988, Verlag, Böhlau) |
Austrian German | External links | External links
Austrian German – German Dictionary
Das Österreichische Volkswörterbuch
Category:Bavarian language
Category:German dialects
German
Category:National varieties of German |
Austrian German | Table of Content | short description, History, Standard Austrian German, Former spoken standard, Special written forms, European Union, Grammar, Verbs, Vocabulary, Dialects, Classification, Regional accents, References, Notes, Citations, Works cited, Further reading, External links |
Axiom of choice | Short description | thumb|250px|Illustration of the axiom of choice, with each set Si represented as a jar and its elements represented as marbles. Each element xi is represented as a marble on the right. Colors are used to suggest a functional association of marbles after adopting the choice axiom. The existence of such a choice function is in general independent of ZF for collections of infinite cardinality, even if all Si are finite.
thumb|250px|(Si) is an infinite indexed family of sets indexed over the real numbers R; that is, there is a set Si for each real number i, with a small sample shown above. Each set contains at least one, and possibly infinitely many, elements. The axiom of choice allows us to select a single element from each set, forming a corresponding family of elements (xi) also indexed over the real numbers, with xi drawn from Si. In general, the collections may be indexed over any set I, (called index set whose elements are used as indices for elements in a set) not just R.
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family of nonempty sets, there exists an indexed set such that for every . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem..
The axiom of choice is equivalent to the statement that every partition has a transversal.
In many cases, a set created by choosing elements can be made without invoking the axiom of choice, particularly if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available — some distinguishing property that happens to hold for exactly one element in each set. An illustrative example is sets picked from the natural numbers. From such sets, one may always select the smallest number, e.g. given the sets {{4, 5, 6}, {10, 12}, {1, 400, 617, 8000}}, the set containing each smallest element is {4, 10, 1}. In this case, "select the smallest number" is a choice function. Even if infinitely many sets are collected from the natural numbers, it will always be possible to choose the smallest element from each set to produce a set. That is, the choice function provides the set of chosen elements. But no definite choice function is known for the collection of all non-empty subsets of the real numbers. In that case, the axiom of choice must be invoked.
Bertrand Russell coined an analogy: for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate collection (i.e. set) of shoes; this makes it possible to define a choice function directly. For an infinite collection of pairs of socks (assumed to have no distinguishing features such as being a left sock rather than a right sock), there is no obvious way to make a function that forms a set out of selecting one sock from each pair without invoking the axiom of choice.
Although originally controversial, the axiom of choice is now used without reservation by most mathematicians,Jech, 1977, p. 348ff; Martin-Löf 2008, p. 210. According to :
The status of the Axiom of Choice has become less controversial in recent years. To most mathematicians it seems quite plausible and it has so many important applications in practically all branches of mathematics that not to accept it would seem to be a wilful hobbling of the practicing mathematician. and is included in the standard form of axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice (ZFC). One motivation for this is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs. Contemporary set theorists also study axioms that are not compatible with the axiom of choice, such as the axiom of determinacy. The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced. |
Axiom of choice | Statement | Statement
A choice function (also called selector or selection) is a function , defined on a collection of nonempty sets, such that for every set in , is an element of . With this concept, the axiom can be stated:
Formally, this may be expressed as follows:
Thus, the negation of the axiom may be expressed as the existence of a collection of nonempty sets which has no choice function. Formally, this may be derived making use of the logical equivalence of
Each choice function on a collection of nonempty sets is an element of the Cartesian product of the sets in . This is not the most general situation of a Cartesian product of a family of sets, where a given set can occur more than once as a factor; however, one can focus on elements of such a product that select the same element every time a given set appears as factor, and such elements correspond to an element of the Cartesian product of all distinct sets in the family. The axiom of choice asserts the existence of such elements; it is therefore equivalent to:
Given any family of nonempty sets, their Cartesian product is a nonempty set. |
Axiom of choice | Nomenclature | Nomenclature
In this article and other discussions of the Axiom of Choice the following abbreviations are common:
AC – the Axiom of Choice. More rarely, AoC is used.
ZF – Zermelo–Fraenkel set theory omitting the Axiom of Choice.
ZFC – Zermelo–Fraenkel set theory, extended to include the Axiom of Choice. |
Axiom of choice | Variants | Variants
There are many other equivalent statements of the axiom of choice. These are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it.
One variation avoids the use of choice functions by, in effect, replacing each choice function with its range:
Given any set , if the empty set is not an element of and the elements of are pairwise disjoint, then there exists a set such that its intersection with any of the elements of contains exactly one element.. According to , this was the formulation of the axiom of choice which was originally given by . See also for this formulation.
This can be formalized in first-order logic as:
Note that is logically equivalent to .
In English, this first-order sentence reads:
Given any set ,
contains the empty set as an element or
the elements of are not pairwise disjoint or
there exists a set such that its intersection with any of the elements of contains exactly one element.
This guarantees for any partition of a set the existence of a subset of containing exactly one element from each part of the partition.
Another equivalent axiom only considers collections that are essentially powersets of other sets:
For any set , the power set of (with the empty set removed) has a choice function.
Authors who use this formulation often speak of the choice function on , but this is a slightly different notion of choice function. Its domain is the power set of (with the empty set removed), and so makes sense for any set , whereas with the definition used elsewhere in this article, the domain of a choice function on a collection of sets is that collection, and so only makes sense for sets of sets. With this alternate notion of choice function, the axiom of choice can be compactly stated as
Every set has a choice function..
which is equivalent to
For any set there is a function such that for any non-empty subset of , lies in .
The negation of the axiom can thus be expressed as:
There is a set such that for all functions (on the set of non-empty subsets of ), there is a subset such that does not lie in . |
Axiom of choice | Restriction to finite sets | Restriction to finite sets
The usual statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite, and thus implies that every finite collection of nonempty sets has a choice function. However, that particular case is a theorem of the Zermelo–Fraenkel set theory without the axiom of choice (ZF); it is easily proved by the principle of finite induction.Tourlakis (2003), pp. 209–210, 215–216. In the even simpler case of a collection of one set, a choice function just corresponds to an element, so this instance of the axiom of choice says that every nonempty set has an element; this holds trivially. The axiom of choice can be seen as asserting the generalization of this property, already evident for finite collections, to arbitrary collections. |
Axiom of choice | Usage | Usage
Until the late 19th century, the axiom of choice was often used implicitly, although it had not yet been formally stated. For example, after having established that the set X contains only non-empty sets, a mathematician might have said "let F(s) be one of the members of s for all s in X" to define a function F. In general, it is impossible to prove that F exists without the axiom of choice, but this seems to have gone unnoticed until Zermelo.
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Axiom of choice | Examples | Examples
The nature of the individual nonempty sets in the collection may make it possible to avoid the axiom of choice even for certain infinite collections. For example, suppose that each member of the collection X is a nonempty subset of the natural numbers. Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set. This gives us a definite choice of an element from each set, and makes it unnecessary to add the axiom of choice to our axioms of set theory.
The difficulty appears when there is no natural choice of elements from each set. If we cannot make explicit choices, how do we know that our selection forms a legitimate set (as defined by the other ZF axioms of set theory)? For example, suppose that X is the set of all non-empty subsets of the real numbers. First we might try to proceed as if X were finite. If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently we shall never be able to produce a choice function for all of X. Next we might try specifying the least element from each set. But some subsets of the real numbers do not have least elements. For example, the open interval (0,1) does not have a least element: if x is in (0,1), then so is x/2, and x/2 is always strictly smaller than x. So this attempt also fails.
Additionally, consider for instance the unit circle S, and the action on S by a group G consisting of all rational rotations, that is, rotations by angles which are rational multiples of π. Here G is countable while S is uncountable. Hence S breaks up into uncountably many orbits under G. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset X of S with the property that all of its translates by G are disjoint from X. The set of those translates partitions the circle into a countable collection of pairwise disjoint sets, which are all pairwise congruent. Since X is not measurable for any rotation-invariant countably additive finite measure on S, finding an algorithm to form a set from selecting a point in each orbit requires that one add the axiom of choice to our axioms of set theory. See non-measurable set for more details.
In classical arithmetic, the natural numbers are well-ordered: for every nonempty subset of the natural numbers, there is a unique least element under the natural ordering. In this way, one may specify a set from any given subset. One might say, "Even though the usual ordering of the real numbers does not work, it may be possible to find a different ordering of the real numbers which is a well-ordering. Then our choice function can choose the least element of every set under our unusual ordering." The problem then becomes that of constructing a well-ordering, which turns out to require the axiom of choice for its existence; every set can be well-ordered if and only if the axiom of choice holds. |
Axiom of choice | Criticism and acceptance | Criticism and acceptance
A proof requiring the axiom of choice may establish the existence of an object without explicitly defining the object in the language of set theory. For example, while the axiom of choice implies that there is a well-ordering of the real numbers, there are models of set theory with the axiom of choice in which no individual well-ordering of the reals is definable. Similarly, although a subset of the real numbers that is not Lebesgue measurable can be proved to exist using the axiom of choice, it is consistent that no such set is definable..
The axiom of choice asserts the existence of these intangibles (objects that are proved to exist, but which cannot be explicitly constructed), which may conflict with some philosophical principles.. Because there is no canonical well-ordering of all sets, a construction that relies on a well-ordering may not produce a canonical result, even if a canonical result is desired (as is often the case in category theory). This has been used as an argument against the use of the axiom of choice.
Another argument against the axiom of choice is that it implies the existence of objects that may seem counterintuitive.. One example is the Banach–Tarski paradox, which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many pieces and, using only rotations and translations, reassemble the pieces into two solid balls each with the same volume as the original. The pieces in this decomposition, constructed using the axiom of choice, are non-measurable sets.
Despite these seemingly paradoxical results, most mathematicians accept the axiom of choice as a valid principle for proving new results in mathematics. But the debate is interesting enough that it is considered notable when a theorem in ZFC (ZF plus AC) is logically equivalent (with just the ZF axioms) to the axiom of choice, and mathematicians look for results that require the axiom of choice to be false, though this type of deduction is less common than the type that requires the axiom of choice to be true.
Theorems of ZF hold true in any model of that theory, regardless of the truth or falsity of the axiom of choice in that particular model. The implications of choice below, including weaker versions of the axiom itself, are listed because they are not theorems of ZF. The Banach–Tarski paradox, for example, is neither provable nor disprovable from ZF alone: it is impossible to construct the required decomposition of the unit ball in ZF, but also impossible to prove there is no such decomposition. Such statements can be rephrased as conditional statements—for example, "If AC holds, then the decomposition in the Banach–Tarski paradox exists." Such conditional statements are provable in ZF when the original statements are provable from ZF and the axiom of choice. |
Axiom of choice | In constructive mathematics | In constructive mathematics
As discussed above, in the classical theory of ZFC, the axiom of choice enables nonconstructive proofs in which the existence of a type of object is proved without an explicit instance being constructed. In fact, in set theory and topos theory, Diaconescu's theorem shows that the axiom of choice implies the law of excluded middle. The principle is thus not available in constructive set theory, where non-classical logic is employed.
The situation is different when the principle is formulated in Martin-Löf type theory. There and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is (depending on approach) included as an axiom or provable as a theorem.Per Martin-Löf, Intuitionistic type theory, 1980.
Anne Sjerp Troelstra, Metamathematical investigation of intuitionistic arithmetic and analysis, Springer, 1973. A cause for this difference is that the axiom of choice in type theory does not have the extensionality properties that the axiom of choice in constructive set theory does. The type theoretical context is discussed further below.
Different choice principles have been thoroughly studied in the constructive contexts and the principles' status varies between different school and varieties of the constructive mathematics.
Some results in constructive set theory use the axiom of countable choice or the axiom of dependent choice, which do not imply the law of the excluded middle. Errett Bishop, who is notable for developing a framework for constructive analysis, argued that an axiom of choice was constructively acceptable, saying
Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.Fred Richman, "Constructive mathematics without choice", in: Reuniting the Antipodes—Constructive and Nonstandard Views of the Continuum (P. Schuster et al., eds), Synthèse Library 306, 199–205, Kluwer Academic Publishers, Amsterdam, 2001. |
Axiom of choice | Independence | Independence
It has been known since as early as 1922 that the axiom of choice may fail in a variant of ZF with urelements, through the technique of permutation models introduced by Abraham Fraenkel and developed further by Andrzej Mostowski. The basic technique can be illustrated as follows: Let xn and yn be distinct urelements for , and build a model where each set is symmetric under the interchange xn ↔ yn for all but a finite number of n. Then the set can be in the model but sets such as cannot, and thus X cannot have a choice function.
In 1938, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model (the constructible universe) that satisfies ZFC, thus showing that ZFC is consistent if ZF itself is consistent. In 1963, Paul Cohen employed the technique of forcing, developed for this purpose, to show that, assuming ZF is consistent, the axiom of choice itself is not a theorem of ZF. He did this by constructing a much more complex model that satisfies ZF¬C (ZF with the negation of AC added as axiom) and thus showing that ZF¬C is consistent. Cohen's model is a symmetric model, which is similar to permutation models, but uses "generic" subsets of the natural numbers (justified by forcing) in place of urelements.
Together these results establish that the axiom of choice is logically independent of ZF. The assumption that ZF is consistent is harmless because adding another axiom to an already inconsistent system cannot make the situation worse. Because of independence, the decision whether to use the axiom of choice (or its negation) in a proof cannot be made by appeal to other axioms of set theory. It must be made on other grounds.
One argument in favor of using the axiom of choice is that it is convenient because it allows one to prove some simplifying propositions that otherwise could not be proved. Many theorems provable using choice are of an elegant general character: the cardinalities of any two sets are comparable, every nontrivial ring with unity has a maximal ideal, every vector space has a basis, every connected graph has a spanning tree, and every product of compact spaces is compact, among many others. Frequently, the axiom of choice allows generalizing a theorem to "larger" objects. For example, it is provable without the axiom of choice that every vector space of finite dimension has a basis, but the generalization to all vector spaces requires the axiom of choice. Likewise, a finite product of compact spaces can be proven to be compact without the axiom of choice, but the generalization to infinite products (Tychonoff's theorem) requires the axiom of choice.
The proof of the independence result also shows that a wide class of mathematical statements, including all statements that can be phrased in the language of Peano arithmetic, are provable in ZF if and only if they are provable in ZFC.This is because arithmetical statements are absolute to the constructible universe L. Shoenfield's absoluteness theorem gives a more general result. Statements in this class include the statement that P = NP, the Riemann hypothesis, and many other unsolved mathematical problems. When attempting to solve problems in this class, it makes no difference whether ZF or ZFC is employed if the only question is the existence of a proof. It is possible, however, that there is a shorter proof of a theorem from ZFC than from ZF.
The axiom of choice is not the only significant statement that is independent of ZF. For example, the generalized continuum hypothesis (GCH) is not only independent of ZF, but also independent of ZFC. However, ZF plus GCH implies AC, making GCH a strictly stronger claim than AC, even though they are both independent of ZF. |
Axiom of choice | Stronger axioms | Stronger axioms
The axiom of constructibility and the generalized continuum hypothesis each imply the axiom of choice and so are strictly stronger than it. In class theories such as Von Neumann–Bernays–Gödel set theory and Morse–Kelley set theory, there is an axiom called the axiom of global choice that is stronger than the axiom of choice for sets because it also applies to proper classes. The axiom of global choice follows from the axiom of limitation of size. Tarski's axiom, which is used in Tarski–Grothendieck set theory and states (in the vernacular) that every set belongs to Grothendieck universe, is stronger than the axiom of choice. |
Axiom of choice | Equivalents | Equivalents
There are important statements that, assuming the axioms of ZF but neither AC nor ¬AC, are equivalent to the axiom of choice.See , for a structured list of 74 equivalents. See , for 86 equivalents with source references. The most important among them are Zorn's lemma and the well-ordering theorem. In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering theorem.
Set theory
Tarski's theorem about choice: For every infinite set A, there is a bijective map between the sets A and A×A.
Trichotomy: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other.
Given two non-empty sets, one has a surjection to the other.
Every surjective function has a right inverse.
The Cartesian product of any family of nonempty sets is nonempty. In other words, every family of nonempty sets has a choice function (i.e. a function which maps each of the nonempty sets to one of its elements).
König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals. (The reason for the term "colloquially" is that the sum or product of a "sequence" of cardinals cannot itself be defined without some aspect of the axiom of choice.)
Well-ordering theorem: Every set can be well-ordered. Consequently, every cardinal has an initial ordinal.
Zorn's lemma: Every non-empty partially ordered set in which every chain (i.e., totally ordered subset) has an upper bound contains at least one maximal element.
Hausdorff maximal principle: Every partially ordered set has a maximal chain. Equivalently, in any partially ordered set, every chain can be extended to a maximal chain.
Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion.
Antichain principle: Every partially ordered set has a maximal antichain. Equivalently, in any partially ordered set, every antichain can be extended to a maximal antichain.
The powerset of any ordinal can be well-ordered.
Abstract algebra
Every vector space has a basis (i.e., a linearly independent spanning subset). In other words, vector spaces are equivalent to free modules.
Krull's theorem: Every unital ring (other than the trivial ring) contains a maximal ideal. Equivalently, in any nontrivial unital ring, every ideal can be extended to a maximal ideal.
For every non-empty set S there is a binary operation defined on S that gives it a group structure.A. Hajnal, A. Kertész: Some new algebraic equivalents of the axiom of choice, Publ. Math. Debrecen, 19(1972), 339–340, see also H. Rubin, J. Rubin, Equivalents of the axiom of choice, II, North-Holland, 1985, p. 111. (A cancellative binary operation is enough, see group structure and the axiom of choice.)
Every free abelian group is projective.
Baer's criterion: Every divisible abelian group is injective.
Every set is a projective object in the category Set of sets.
Functional analysis
The closed unit ball of the dual of a normed vector space over the reals has an extreme point.
Point-set topology
The Cartesian product of any family of connected topological spaces is connected.
Tychonoff's theorem: The Cartesian product of any family of compact topological spaces is compact.
In the product topology, the closure of a product of subsets is equal to the product of the closures.
Mathematical logic
If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem; see the section "Weaker forms" below.
Lowenheim-Skolem theorem: If first-order theory has infinite model, then it has infinite model of every possible cardinality greater than cardinality of language of this theory.
Graph theory
Every connected graph has a spanning tree. Equivalently, every nonempty graph has a spanning forest.; . See in particular Theorem 2.1, pp. 192–193. |
Axiom of choice | Category theory | Category theory
Several results in category theory invoke the axiom of choice for their proof. These results might be weaker than, equivalent to, or stronger than the axiom of choice, depending on the strength of the technical foundations. For example, if one defines categories in terms of sets, that is, as sets of objects and morphisms (usually called a small category), then there is no category of all sets, and so it is difficult for a category-theoretic formulation to apply to all sets. On the other hand, other foundational descriptions of category theory are considerably stronger, and an identical category-theoretic statement of choice may be stronger than the standard formulation, à la class theory, mentioned above.
Examples of category-theoretic statements which require choice include:
Every small category has a skeleton.
If two small categories are weakly equivalent, then they are equivalent.
Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint (the Freyd adjoint functor theorem). |
Axiom of choice | Weaker forms | Weaker forms
There are several weaker statements that are not equivalent to the axiom of choice but are closely related. One example is the axiom of dependent choice (DC). A still weaker example is the axiom of countable choice (ACω or CC), which states that a choice function exists for any countable set of nonempty sets. These axioms are sufficient for many proofs in elementary mathematical analysis, and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are disprovable from the full axiom of choice.
Given an ordinal parameter α ≥ ω+2 — for every set S with rank less than α, S is well-orderable. Given an ordinal parameter α ≥ 1 — for every set S with Hartogs number less than ωα, S is well-orderable. As the ordinal parameter is increased, these approximate the full axiom of choice more and more closely.
Other choice axioms weaker than axiom of choice include the Boolean prime ideal theorem and the axiom of uniformization. The former is equivalent in ZF to Tarski's 1930 ultrafilter lemma: every filter is a subset of some ultrafilter. |
Axiom of choice | Results requiring AC (or weaker forms) but weaker than it | Results requiring AC (or weaker forms) but weaker than it
One of the most interesting aspects of the axiom of choice is the large number of places in mathematics where it shows up. Here are some statements that require the axiom of choice in the sense that they are not provable from ZF but are provable from ZFC (ZF plus AC). Equivalently, these statements are true in all models of ZFC but false in some models of ZF.
Set theory
The ultrafilter lemma (with ZF) can be used to prove the Axiom of choice for finite sets: Given and a collection of non-empty sets, their product is not empty.
The union of any countable family of countable sets is countable (this requires countable choice but not the full axiom of choice).
If the set A is infinite, then there exists an injection from the natural numbers N to A (see Dedekind infinite).It is shown by , that the axiom of countable choice implies the equivalence of infinite and Dedekind-infinite sets, but that the equivalence of infinite and Dedekind-infinite sets does not imply the axiom of countable choice in ZF.
Eight definitions of a finite set are equivalent.It was shown by and others using Mostowski models that eight definitions of a finite set are independent in ZF without AC, although they are equivalent when AC is assumed. The definitions are I-finite, Ia-finite, II-finite, III-finite, IV-finite, V-finite, VI-finite and VII-finite. I-finiteness is the same as normal finiteness. IV-finiteness is the same as Dedekind-finiteness.
Every infinite game in which is a Borel subset of Baire space is determined.
Every infinite cardinal κ satisfies 2×κ = κ.
Measure theory
The Vitali theorem on the existence of non-measurable sets, which states that there exists a subset of the real numbers that is not Lebesgue measurable.
There exist Lebesgue-measurable subsets of the real numbers that are not Borel sets. That is, the Borel σ-algebra on the real numbers (which is generated by all real intervals) is strictly included the Lebesgue-measure σ-algebra on the real numbers.
The Hausdorff paradox.
The Banach–Tarski paradox.
Algebra
Every field has an algebraic closure.
Every field extension has a transcendence basis.
Every infinite-dimensional vector space contains an infinite linearly independent subset (this requires dependent choice, but not the full axiom of choice).
Stone's representation theorem for Boolean algebras needs the Boolean prime ideal theorem.
The Nielsen–Schreier theorem, that every subgroup of a free group is free.
The additive groups of R and C are isomorphic.
Functional analysis
The Hahn–Banach theorem in functional analysis, allowing the extension of linear functionals.
The theorem that every Hilbert space has an orthonormal basis.
The Banach–Alaoglu theorem about compactness of sets of functionals.
The Baire category theorem about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem.
On every infinite-dimensional topological vector space there is a discontinuous linear map.
General topology
A uniform space is compact if and only if it is complete and totally bounded.
Every Tychonoff space has a Stone–Čech compactification.
Mathematical logic
Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion. That is, every consistent set of first-order sentences can be extended to a maximal consistent set.
The compactness theorem: If is a set of first-order (or alternatively, zero-order) sentences such that every finite subset of has a model, then has a model. |
Axiom of choice | Possibly equivalent implications of AC | Possibly equivalent implications of AC
There are several historically important set-theoretic statements implied by AC whose equivalence to AC is open. Zermelo cited the partition principle, which was formulated before AC itself, as a justification for believing AC. In 1906, Russell declared PP to be equivalent, but whether the partition principle implies AC is the oldest open problem in set theory, and the equivalences of the other statements are similarly hard old open problems. In every known model of ZF where choice fails, these statements fail too, but it is unknown whether they can hold without choice.
Set theory
Partition principle: if there is a surjection from A to B, there is an injection from B to A. Equivalently, every partition P of a set S is less than or equal to S in size.
Converse Schröder–Bernstein theorem: if two sets have surjections to each other, they are equinumerous.
Weak partition principle: if there is an injection and a surjection from A to B, then A and B are equinumerous. Equivalently, a partition of a set S cannot be strictly larger than S. If WPP holds, this already implies the existence of a non-measurable set. Each of the previous three statements is implied by the preceding one, but it is unknown if any of these implications can be reversed.
There is no infinite decreasing sequence of cardinals. The equivalence was conjectured by Schoenflies in 1905.
Abstract algebra
Hahn embedding theorem: Every ordered abelian group G order-embeds as a subgroup of the additive group endowed with a lexicographical order, where Ω is the set of Archimedean equivalence classes of G. This equivalence was conjectured by Hahn in 1907. |
Axiom of choice | Stronger forms of the negation of AC | Stronger forms of the negation of AC
If we abbreviate by BP the claim that every set of real numbers has the property of Baire, then BP is stronger than ¬AC, which asserts the nonexistence of any choice function on perhaps only a single set of nonempty sets. Strengthened negations may be compatible with weakened forms of AC. For example, ZF + DCAxiom of dependent choice + BP is consistent, if ZF is.
It is also consistent with ZF + DC that every set of reals is Lebesgue measurable, but this consistency result, due to Robert M. Solovay, cannot be proved in ZFC itself, but requires a mild large cardinal assumption (the existence of an inaccessible cardinal). The much stronger axiom of determinacy, or AD, implies that every set of reals is Lebesgue measurable, has the property of Baire, and has the perfect set property (all three of these results are refuted by AC itself). ZF + DC + AD is consistent provided that a sufficiently strong large cardinal axiom is consistent (the existence of infinitely many Woodin cardinals).
Quine's system of axiomatic set theory, New Foundations (NF), takes its name from the title ("New Foundations for Mathematical Logic") of the 1937 article that introduced it. In the NF axiomatic system, the axiom of choice can be disproved. |
Axiom of choice | Statements implying the negation of AC | Statements implying the negation of AC
There are models of Zermelo-Fraenkel set theory in which the axiom of choice is false. We shall abbreviate "Zermelo-Fraenkel set theory plus the negation of the axiom of choice" by ZF¬C. For certain models of ZF¬C, it is possible to validate the negation of some standard ZFC theorems. As any model of ZF¬C is also a model of ZF, it is the case that for each of the following statements, there exists a model of ZF in which that statement is true.
The negation of the weak partition principle: There is a set that can be partitioned into strictly more equivalence classes than the original set has elements, and a function whose domain is strictly smaller than its range. In fact, this is the case in all known models.
There is a function f from the real numbers to the real numbers such that f is not continuous at a, but f is sequentially continuous at a, i.e., for any sequence {xn} converging to a, limn f(xn)=f(a).
There is an infinite set of real numbers without a countably infinite subset.
The real numbers are a countable union of countable sets., Theorem 10.6 with proof. This does not imply that the real numbers are countable: As pointed out above, to show that a countable union of countable sets is itself countable requires the Axiom of countable choice.
There is a field with no algebraic closure.
In all models of ZF¬C there is a vector space with no basis.
There is a vector space with two bases of different cardinalities.
There is a free complete Boolean algebra on countably many generators.
There is a set that cannot be linearly ordered.
There exists a model of ZF¬C in which every set in Rn is measurable. Thus it is possible to exclude counterintuitive results like the Banach–Tarski paradox which are provable in ZFC. Furthermore, this is possible whilst assuming the Axiom of dependent choice, which is weaker than AC but sufficient to develop most of real analysis.
In all models of ZF¬C, the generalized continuum hypothesis does not hold.
For proofs, see .
Additionally, by imposing definability conditions on sets (in the sense of descriptive set theory) one can often prove restricted versions of the axiom of choice from axioms incompatible with general choice. This appears, for example, in the Moschovakis coding lemma. |
Axiom of choice | Axiom of choice in type theory | Axiom of choice in type theory
In type theory, a different kind of statement is known as the axiom of choice. This form begins with two types, σ and τ, and a relation R between objects of type σ and objects of type τ. The axiom of choice states that if for each x of type σ there exists a y of type τ such that R(x,y), then there is a function f from objects of type σ to objects of type τ such that R(x,f(x)) holds for all x of type σ:
Unlike in set theory, the axiom of choice in type theory is typically stated as an axiom scheme, in which R varies over all formulas or over all formulas of a particular logical form. |
Axiom of choice | Notes | Notes |
Axiom of choice | References | References
Per Martin-Löf, "100 years of Zermelo's axiom of choice: What was the problem with it?", in Logicism, Intuitionism, and Formalism: What Has Become of Them?, Sten Lindström, Erik Palmgren, Krister Segerberg, and Viggo Stoltenberg-Hansen, editors (2008).
, available as a Dover Publications reprint, 2013, .
Herman Rubin, Jean E. Rubin: Equivalents of the axiom of choice. North Holland, 1963. Reissued by Elsevier, April 1970. .
Herman Rubin, Jean E. Rubin: Equivalents of the Axiom of Choice II. North Holland/Elsevier, July 1985, .
George Tourlakis, Lectures in Logic and Set Theory. Vol. II: Set Theory, Cambridge University Press, 2003.
Ernst Zermelo, "Untersuchungen über die Grundlagen der Mengenlehre I," Mathematische Annalen 65: (1908) pp. 261–81. PDF download via digizeitschriften.de
Translated in: Jean van Heijenoort, 2002. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. New edition. Harvard University Press.
1904. "Proof that every set can be well-ordered," 139-41.
1908. "Investigations in the foundations of set theory I," 199–215. |
Axiom of choice | External links | External links
Axiom of Choice entry in the Springer Encyclopedia of Mathematics.
Axiom of Choice and Its Equivalents entry at ProvenMath. Includes formal statement of the Axiom of Choice, Hausdorff's Maximal Principle, Zorn's Lemma and formal proofs of their equivalence down to the finest detail.
Consequences of the Axiom of Choice , based on the book by Paul Howard and Jean Rubin.
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Axiom of choice | Table of Content | Short description, Statement, Nomenclature, Variants, Restriction to finite sets, Usage, Examples, Criticism and acceptance, In constructive mathematics, Independence, Stronger axioms, Equivalents, Category theory, Weaker forms, Results requiring AC (or weaker forms) but weaker than it, Possibly equivalent implications of AC, Stronger forms of the negation of AC, Statements implying the negation of AC, Axiom of choice in type theory, Notes, References, External links |
Attila | redirect2 | Attila ( or ; ), frequently called Attila the Hun, was the ruler of the Huns from 434 until his death in early 453. He was also the leader of an empire consisting of Huns, Ostrogoths, Alans, and Gepids, among others, in Central and Eastern Europe.
As nephews to Rugila, Attila and his elder brother Bleda succeeded him to the throne in 435, ruling jointly until the death of Bleda in 445. During his reign, Attila was one of the most feared enemies of the Western and Eastern Roman Empires. He crossed the Danube twice and plundered the Balkans but was unable to take Constantinople. In 441, he led an invasion of the Eastern Roman (Byzantine) Empire, the success of which emboldened him to invade the West. He also attempted to conquer Roman Gaul (modern France), crossing the Rhine in 451 and marching as far as Aurelianum (Orléans), before being stopped in the Battle of the Catalaunian Plains.
He subsequently invaded Italy, devastating the northern provinces, but was unable to take Rome. He planned for further campaigns against the Romans but died in 453. After Attila's death, his close adviser, Ardaric of the Gepids, led a Germanic revolt against Hunnic rule, after which the Hunnic Empire quickly collapsed. Attila lived on as a character in Germanic heroic legend. |
Attila | Etymology | Etymology
Many scholars have argued that the name Attila derives from East Germanic origin; Attila is formed from the Gothic or Gepidic noun atta, "father", by means of the diminutive suffix -ila, meaning "little father", compare Wulfila from wulfs "wolf" and -ila, i.e. "little wolf". The Gothic etymology was first proposed by Jacob and Wilhelm Grimm in the early 19th century. Maenchen-Helfen notes that this derivation of the name "offers neither phonetic nor semantic difficulties", and Gerhard Doerfer notes that the name is simply correct Gothic. Alexander Savelyev and Choongwon Jeong (2020) similarly state that Attila's name "must have been Gothic in origin." The name has sometimes been interpreted as a Germanization of a name of Hunnic origin.
Other scholars have argued for a Turkic origin of the name. Omeljan Pritsak considered Ἀττίλα (Attíla) a composite title-name which derived from Turkic *es (great, old), and *til (sea, ocean), and the suffix /a/. The stressed back syllabic til assimilated the front member es, so it became *as. It is a nominative, in form of attíl- (< *etsíl < *es tíl) with the meaning "the oceanic, universal ruler". J. J. Mikkola connected it with Turkic āt (name, fame). As another Turkic possibility, H. Althof (1902) considered it was related to Turkish atli (horseman, cavalier), or Turkish at (horse) and dil (tongue). Maenchen-Helfen argues that Pritsak's derivation is "ingenious but for many reasons unacceptable", while dismissing Mikkola's as "too farfetched to be taken seriously". M. Snædal similarly notes that none of these proposals has achieved wide acceptance.
Criticizing the proposals of finding Turkic or other etymologies for Attila, Doerfer notes that King George VI of the United Kingdom had a name of Greek origin, and Süleyman the Magnificent had a name of Arabic origin, yet that does not make them Greek or Arab: it is therefore plausible that Attila would have a name not of Hunnic origin. Historian Hyun Jin Kim, however, has argued that the Turkic etymology is "more probable".
M. Snædal, in a paper that rejects the Germanic derivation but notes the problems with the existing proposed Turkic etymologies, argues that Attila's name could have originated from Turkic-Mongolian at, adyy/agta (gelding, warhorse) and Turkish atlı (horseman, cavalier), meaning "possessor of geldings, provider of warhorses". |
Attila | Historiography and sources | Historiography and sources
thumb|Figure of Attila in a museum in Hungary.
thumb|A reconstruction of Attila by George S. Stuart, Museum of Ventura County, USA.
thumb|Mór Than's 19th century painting of The Feast of Attila, based on a fragment of Priscus.
The historiography of Attila is faced with a major challenge, in that the only complete sources are written in Greek and Latin by the enemies of the Huns. Attila's contemporaries left many testimonials of his life, but only fragments of these remain. Priscus was a Byzantine diplomat and historian who wrote in Greek, and he was both a witness to and an actor in the story of Attila, as a member of the embassy of Theodosius II at the Hunnic court in 449. He was obviously biased by his political position, but his writing is a major source for information on the life of Attila, and he is the only person known to have recorded a physical description of him. He wrote a history of the late Roman Empire in eight books covering the period from 430 to 476.
Only fragments of Priscus' work remain. It was cited extensively by 6th-century historians Procopius and Jordanes, especially in Jordanes' The Origin and Deeds of the Goths, which contains numerous references to Priscus's history, and it is also an important source of information about the Hunnic empire and its neighbors. He describes the legacy of Attila and the Hunnic people for a century after Attila's death. Marcellinus Comes, a chancellor of Justinian during the same era, also describes the relations between the Huns and the Eastern Roman Empire.
Numerous ecclesiastical writings contain useful but scattered information, sometimes difficult to authenticate or distorted by years of hand-copying between the 6th and 17th centuries. The Hungarian writers of the 12th century wished to portray the Huns in a positive light as their glorious ancestors, and so repressed certain historical elements and added their own legends.
The literature and knowledge of the Huns themselves was transmitted orally, by means of epics and chanted poems that were handed down from generation to generation. Indirectly, fragments of this oral history have reached us via the literature of the Scandinavians and Germans, neighbors of the Huns who wrote between the 9th and 13th centuries. Attila is a major character in many Medieval epics, such as the Nibelungenlied, as well as various Eddas and sagas.
Archaeological investigation has uncovered some details about the lifestyle, art, and warfare of the Huns. There are a few traces of battles and sieges, but the tomb of Attila and the location of his capital have not yet been found. |
Attila | Appearance and character | Appearance and character
There is no surviving first-hand account of Attila's appearance, but there is a possible second-hand source provided by Jordanes, who cites a description given by Priscus.
Some scholars have suggested that these features are typically East Asian, because in combination they fit the physical type of people from Eastern Asia, so Attila's ancestors may have come from there. Other historians have suggested that the same features may have been typical of some Scythian people.Wolff, Larry. Inventing Eastern Europe: The Map of Civilization on the Mind of the Enlightenment. Stanford University Press; (1994). pp. 299–230. Fields, Nic. Attila the Hun (Command). Osprey Publishing; UK ed. (2015). pp. 58–60. |
Attila | Early life and background | Early life and background
thumb|left|Huns in battle with the Alans. An 1870s engraving after a drawing by Johann Nepomuk Geiger (1805–1880).
The Huns were a group of Eurasian nomads, appearing from east of the Volga, who migrated further into Western Europe c. 370 and built up an enormous empire there. Their main military techniques were mounted archery and javelin throwing. They were in the process of developing settlements before their arrival in Western Europe, yet the Huns were a society of pastoral warriors whose primary form of nourishment was meat and milk, products of their herds.
The origin and language of the Huns has been the subject of debate for centuries. According to some theories, their leaders at least may have spoken a Turkic language, perhaps closest to the modern Chuvash language. According to the Encyclopedia of European Peoples, "the Huns, especially those who migrated to the west, may have been a combination of central Asian Turkic, Mongolic, and Ugric stocks".
Attila's father Mundzuk was the brother of kings Octar and Ruga, who reigned jointly over the Hunnic empire in the early fifth century. This form of diarchy was recurrent with the Huns, but historians are unsure whether it was institutionalized, merely customary, or an occasional occurrence. His family was from a noble lineage, but it is uncertain whether they constituted a royal dynasty. Attila's birthdate is debated; journalist Éric Deschodt and writer Herman Schreiber have proposed a date of 395. However, historian Iaroslav Lebedynsky and archaeologist Katalin Escher prefer an estimate between the 390s and the first decade of the fifth century. Several historians have proposed 406 as the date.
Attila grew up in a rapidly changing world. His people were nomads who had only recently arrived in Europe. They crossed the Volga river during the 370s and annexed the territory of the Alans, then attacked the Gothic kingdom between the Carpathian Mountains and the Danube. They were a very mobile people, whose mounted archers had acquired a reputation for invincibility, and the Germanic tribes seemed unable to withstand them. Vast populations fleeing the Huns moved from Germania into the Roman Empire in the west and south, and along the banks of the Rhine and Danube. In 376, the Goths crossed the Danube, initially submitting to the Romans but soon rebelling against Emperor Valens, whom they killed in the Battle of Adrianople in 378. Large numbers of Vandals, Alans, Suebi, and Burgundians crossed the Rhine and invaded Roman Gaul on December 31, 406, to escape the Huns. The Roman Empire had been split in half since 395 and was ruled by two distinct governments, one based in Ravenna in the West, and the other in Constantinople in the East. The Roman Emperors, both East and West, were generally from the Theodosian family in Attila's lifetime (despite several power struggles).
The Huns dominated a vast territory with nebulous borders determined by the will of a constellation of ethnically varied peoples. Some were assimilated to Hunnic nationality, whereas many retained their own identities and rulers but acknowledged the suzerainty of the king of the Huns. The Huns were also the indirect source of many of the Romans' problems, driving various Germanic tribes into Roman territory, yet relations between the two empires were cordial: the Romans used the Huns as mercenaries against the Germans and even in their civil wars. Thus, the usurper Joannes was able to recruit thousands of Huns for his army against Valentinian III in 424. It was Aëtius, later Patrician of the West, who managed this operation. They exchanged ambassadors and hostages, the alliance lasting from 401 to 450 and permitting the Romans numerous military victories. The Huns considered the Romans to be paying them tribute, whereas the Romans preferred to view this as payment for services rendered. The Huns had become a great power by the time that Attila came of age during the reign of his uncle Ruga, to the point that Nestorius, the Patriarch of Constantinople, deplored the situation with these words: "They have become both masters and slaves of the Romans". |
Attila | Campaigns against the Eastern Roman Empire | Campaigns against the Eastern Roman Empire
thumb|right|The Empire of the Huns and subject tribes at the time of Attila.
The death of Rugila (also known as Rua or Ruga) in 434 left the sons of his brother Mundzuk, Attila and Bleda, in control of the united Hun tribes. At the time of the two brothers' accession, the Hun tribes were bargaining with Eastern Roman Emperor Theodosius II's envoys for the return of several renegades who had taken refuge within the Eastern Roman Empire, possibly Hunnic nobles who disagreed with the brothers' assumption of leadership.
The following year, Attila and Bleda met with the imperial legation at Margus (Požarevac), all seated on horseback in the Hunnic manner, and negotiated an advantageous treaty. The Romans agreed to return the fugitives, to double their previous tribute of 350 Roman pounds (c. 115 kg) of gold, to open their markets to Hunnish traders, and to pay a ransom of eight solidi for each Roman taken prisoner by the Huns. The Huns, satisfied with the treaty, decamped from the Roman Empire and returned to their home in the Great Hungarian Plain, perhaps to consolidate and strengthen their empire. Theodosius used this opportunity to strengthen the walls of Constantinople, building the city's first sea wall, and to build up his border defenses along the Danube.
The Huns remained out of Roman sight for the next few years while they invaded the Sassanid Empire. They were defeated in Armenia by the Sassanids, abandoned their invasion, and turned their attentions back to Europe. In 440, they reappeared in force on the borders of the Roman Empire, attacking the merchants at the market on the north bank of the Danube that had been established by the treaty of 435.
Crossing the Danube, they laid waste to the cities of Illyricum and forts on the river, including (according to Priscus) Viminacium, a city of Moesia. Their advance began at Margus, where they demanded that the Romans turn over a bishop who had retained property that Attila regarded as his. While the Romans discussed the bishop's fate, he slipped away secretly to the Huns and betrayed the city to them.
While the Huns attacked city-states along the Danube, the Vandals (led by Geiseric) captured the Western Roman province of Africa and its capital of Carthage. Africa was the richest province of the Western Empire and a main source of food for Rome. The Sassanid Shah Yazdegerd II invaded Armenia in 441.
The Romans stripped the Balkan area of forces, sending them to Sicily in order to mount an expedition against the Vandals in Africa. This left Attila and Bleda a clear path through Illyricum into the Balkans, which they invaded in 441. The Hunnish army sacked Margus and Viminacium, and then took Singidunum (Belgrade) and Sirmium. During 442, Theodosius recalled his troops from Sicily and ordered a large issue of new coins to finance operations against the Huns. He believed that he could defeat the Huns and refused the Hunnish kings' demands.
Attila responded with a campaign in 443. For the first time (as far as the Romans knew) his forces were equipped with battering rams and rolling siege towers, with which they successfully assaulted the military centers of Ratiara and Naissus (Niš) and massacred the inhabitants. Priscus said "When we arrived at Naissus we found the city deserted, as though it had been sacked; only a few sick persons lay in the churches. We halted at a short distance from the river, in an open space, for all the ground adjacent to the bank was full of the bones of men slain in war."
Advancing along the Nišava River, the Huns next took Serdica (Sofia), Philippopolis (Plovdiv), and Arcadiopolis (Lüleburgaz). They encountered and destroyed a Roman army outside Constantinople but were stopped by the double walls of the Eastern capital. They defeated a second army near Callipolis (Gelibolu).
Theodosius, unable to make effective armed resistance, admitted defeat, sending the Magister militum per Orientem Anatolius to negotiate peace terms. The terms were harsher than the previous treaty: the Emperor agreed to hand over 6,000 Roman pounds (c. 2000 kg) of gold as punishment for having disobeyed the terms of the treaty during the invasion; the yearly tribute was tripled, rising to 2,100 Roman pounds (c. 700 kg) in gold; and the ransom for each Roman prisoner rose to 12 solidi.
Their demands were met for a time, and the Hun kings withdrew into the interior of their empire. Bleda died following the Huns' withdrawal from Byzantium (probably around 445). Attila then took the throne for himself, becoming the sole ruler of the Huns. |
Attila | Solitary kingship | Solitary kingship
In 447, Attila again rode south into the Eastern Roman Empire through Moesia. The Roman army, under Gothic magister militum Arnegisclus, met him in the Battle of the Utus and was defeated, though not without inflicting heavy losses. The Huns were left unopposed and rampaged through the Balkans as far as Thermopylae.
Constantinople itself was saved by the Isaurian troops of magister militum per Orientem Zeno and protected by the intervention of prefect Constantinus, who organized the reconstruction of the walls that had been previously damaged by earthquakes and, in some places, to construct a new line of fortification in front of the old. Callinicus, in his Life of Saint Hypatius, wrote: |
Attila | In the west | In the west
thumb|left|The general path of the Hun forces in the invasion of Gaul.
In 450, Attila proclaimed his intent to attack the Visigoth kingdom of Toulouse by making an alliance with Emperor Valentinian III. He had previously been on good terms with the Western Roman Empire and its influential general Flavius Aëtius. Aëtius had spent a brief exile among the Huns in 433, and the troops that Attila provided against the Goths and Bagaudae had helped earn him the largely honorary title of magister militum in the west. The gifts and diplomatic efforts of Geiseric, who opposed and feared the Visigoths, may also have influenced Attila's plans.
However, Valentinian's sister was Honoria, who had sent the Hunnish king a plea for help—and her engagement ring—in order to escape her forced betrothal to a Roman senator in the spring of 450. Honoria may not have intended a proposal of marriage, but Attila chose to interpret her message as such. He accepted, asking for half of the western Empire as dowry.
When Valentinian discovered the plan, only the influence of his mother Galla Placidia convinced him to exile Honoria, rather than killing her. He also wrote to Attila, strenuously denying the legitimacy of the supposed marriage proposal. Attila sent an emissary to Ravenna to proclaim that Honoria was innocent, that the proposal had been legitimate, and that he would come to claim what was rightfully his.
Attila interfered in a succession struggle after the death of a Frankish ruler. Attila supported the elder son, while Aëtius supported the younger. (The location and identity of these kings is not known and subject to conjecture.) Attila gathered his vassals—Gepids, Ostrogoths, Rugians, Scirians, Heruls, Thuringians, Alans, Burgundians, among others—and began his march west. In 451, he arrived in Belgica with an army exaggerated by Jordanes to half a million strong.
thumb|Roman villa in Gaul sacked by Attila's hordes, by French historial painter Georges Rochegrosse
On April 7, he captured Metz. He also captured Strasbourg. Other cities attacked can be determined by the hagiographic vitae written to commemorate their bishops: Nicasius was slaughtered before the altar of his church in Rheims; Servatus is alleged to have saved Tongeren with his prayers, as Saint Genevieve is said to have saved Paris. Lupus, bishop of Troyes, is also credited with saving his city by meeting Attila in person.
Aëtius moved to oppose Attila, gathering troops from among the Franks, the Burgundians, and the Celts. A mission by Avitus and Attila's continued westward advance convinced the Visigoth king Theodoric I (Theodorid) to ally with the Romans. The combined armies reached Orléans ahead of Attila, thus checking and turning back the Hunnish advance. Aëtius gave chase and caught the Huns at a place usually assumed to be near Catalaunum (modern Châlons-en-Champagne). Attila decided to fight the Romans on plains where he could use his cavalry.
The two armies clashed in the Battle of the Catalaunian Plains, the outcome of which is commonly considered to be a strategic victory for the Visigothic-Roman alliance. Theodoric was killed in the fighting, and Aëtius failed to press his advantage, according to Edward Gibbon and Edward Creasy, because he feared the consequences of an overwhelming Visigothic triumph as much as he did a defeat. From Aëtius' point of view, the best outcome was what occurred: Theodoric died, Attila was in retreat and disarray, and the Romans had the benefit of appearing victorious. |
Attila | Invasion of Italy and death | Invasion of Italy and death
thumb|Attila is besieging Aquileia (Chronicon Pictum, 1358).
thumb|Raphael's The Meeting between Leo the Great and Attila depicts Leo, escorted by Saint Peter and Saint Paul, meeting with the Hun emperor outside Rome.
Attila returned in 452 to renew his marriage claim with Honoria, invading and ravaging Italy along the way. Communities became established in what would later become Venice as a result of these attacks when the residents fled to small islands in the Venetian Lagoon. His army sacked numerous cities and razed Aquileia so completely that it was afterwards hard to recognize its original site. Aëtius lacked the strength to offer battle, but managed to harass and slow Attila's advance with only a shadow force. Attila finally halted at the River Po. By this point, disease and starvation may have taken hold in Attila's camp, thus hindering his war efforts and potentially contributing to the cessation of invasion.
Emperor Valentinian III sent three envoys, the high civilian officers Gennadius Avienus and Trigetius, as well as Pope Leo I, who met Attila at Mincio in the vicinity of Mantua and obtained from him the promise that he would withdraw from Italy and negotiate peace with the Emperor. Prosper of Aquitaine gives a short description of the historic meeting, but gives all the credit to Leo for the successful negotiation. Priscus reports that superstitious fear of the fate of Alaric gave him pause—as Alaric died shortly after sacking Rome in 410.
Italy had suffered from a terrible famine in 451 and her crops were faring little better in 452. Attila's devastating invasion of the plains of northern Italy this year did not improve the harvest. To advance on Rome would have required supplies which were not available in Italy, and taking the city would not have improved Attila's supply situation. Therefore, it was more profitable for Attila to conclude peace and retreat to his homeland.
Furthermore, an East Roman force had crossed the Danube under the command of another officer also named Aetius—who had participated in the Council of Chalcedon the previous year—and proceeded to defeat the Huns who had been left behind by Attila to safeguard their home territories. Attila, hence, faced heavy human and natural pressures to retire "from Italy without ever setting foot south of the Po". As Hydatius writes in his Chronica Minora: |
Attila | Death | Death
thumb|The Huns, led by Attila, invade Italy (Attila, the Scourge of God, by Ulpiano Checa, 1887).
In the Eastern Roman Empire, Emperor Marcian succeeded Theodosius II, and stopped paying tribute to the Huns. Attila withdrew from Italy to his palace across the Danube, while making plans to strike at Constantinople once more to reclaim tribute.Kershaw, Stephen P. (2013). A Brief History of the Roman Empire: Rise and Fall. London. Constable & Robinson Ltd. pp. 398, 402–403. .
However, he died in the early months of 453.
The conventional account from Priscus says that Attila was at a feast celebrating his latest marriage, this time to the beautiful young Ildico (the name suggests Gothic or Ostrogoth origins). In the midst of the revels, however, he suffered severe bleeding and died. He may have had a nosebleed and choked to death in a stupor. Or he may have succumbed to internal bleeding, possibly due to ruptured esophageal varices. Esophageal varices are dilated veins that form in the lower part of the esophagus, often caused by years of excessive alcohol consumption; they are fragile and can easily rupture, leading to death by hemorrhage.
Another account of his death was first recorded 80 years after the events by Roman chronicler Marcellinus Comes. It reports that "Attila, King of the Huns and ravager of the provinces of Europe, was pierced by the hand and blade of his wife". One modern analyst suggests that he was assassinated, but most reject these accounts as no more than hearsay, preferring instead the account given by Attila's contemporary Priscus, recounted in the 6th century by Jordanes: |
Attila | Descendants | Descendants
Attila's sons Ellac, Dengizich and Ernak, "in their rash eagerness to rule they all alike destroyed his empire". They "were clamoring that the nations should be divided among them equally and that warlike kings with their peoples should be apportioned to them by lot like a family estate". Against the treatment as "slaves of the basest condition" a Germanic alliance led by the Gepid ruler Ardaric (who was noted for great loyalty to Attila) revolted and fought with the Huns in Pannonia in the Battle of Nedao 454 AD. Attila's eldest son Ellac was killed in that battle. Attila's sons "regarding the Goths as deserters from their rule, came against them as though they were seeking fugitive slaves", attacked Ostrogothic co-ruler Valamir (who also fought alongside Ardaric and Attila at the Catalaunian Plains), but were repelled, and some group of Huns moved to Scythia (probably those of Ernak). His brother Dengizich attempted a renewed invasion across the Danube in 468 AD, but was defeated at the Battle of Bassianae by the Ostrogoths. Dengizich was killed by Roman-Gothic general Anagast the following year, after which the Hunnic dominion ended.
Many of Attila's close relatives are known by name, and some even by deeds, but valid genealogical sources are rare, and there seems to be no verifiable way to trace Attila's descendants beyond a few generations. This has not stopped many genealogists from attempting to reconstruct a valid line of descent to various medieval rulers. One of the most credible claims has been that of the Nominalia of the Bulgarian khans for mythological Avitohol and Irnik from the Dulo clan of the Bulgars. The Hungarian Árpád dynasty also claimed to be a direct descendant of Attila. Medieval Hungarian chronicles from the Hungarian royal court like Gesta Hungarorum, Gesta Hunnorum et Hungarorum, Chronicon Pictum, Buda Chronicle, Chronica Hungarorum claimed that the Árpád dynasty and the Aba clan are the descendants of Attila. |
Attila | Later folklore and iconography | Later folklore and iconography
The name has many variants in several languages: Atli and Atle in Old Norse; Etzel in Middle High German (Nibelungenlied); Ætla in Old English; Attila, Atilla, and Etele in Hungarian (Attila is the most popular); Attila, Atilla, Atilay, or Atila in Turkish; and Adil and Edil in Kazakh or Adil ("same/similar") or Edil ("to use") in Mongolian. |
Attila | Attila and Hun tradition in the medieval Hungarian Royal Court | Attila and Hun tradition in the medieval Hungarian Royal Court
thumb|King Attila on the throne (Chronicon Pictum, 1358).
The basic premise of the Hungarian medieval chronicle tradition that the Huns, i.e. the Hungarians coming out twice from Scythia, the guiding principle of the chronicles was the Hun-Hungarian continuity. The Hungarian state founder royal dynasty, the Árpád dynasty claimed to be a direct descendant of the great Hun leader Attila. Medieval Hungarian chronicles claimed that Grand Prince Árpád of Hungary was the descendant of Attila.
Árpád, Grand Prince of the Hungarians says in the Gesta Hungarorum:
King Matthias of Hungary (1458–1490) was happy to be described as "the second Attila". The Chronica Hungarorum by Johannes Thuróczy set the goal of glorifying Attila, which was undeservedly neglected, moreover, he introduced the famous "Scourge of God" characterization to the later Hungarian writers, because the earlier chronicles remained hidden for a long time. Thuróczy worked hard to endear Attila, the Hun king with an effort far surpassing his predecessor chroniclers. He made Attila a model for his victorious ruler, King Matthias of Hungary who had Attila's abilities, with this he almost brought "the hammer of the world" to life. |
Attila | Legends about Attila and the sword of Mars | Legends about Attila and the sword of Mars
Jordanes embellished the report of Priscus, reporting that Attila had possessed the "Holy War Sword of the Scythians", which was given to him by Mars and made him a "prince of the entire world".
Lampert of Hersfeld's contemporary chronicles report that shortly before the year 1071, the Sword of Attila had been presented to Otto of Nordheim by the exiled queen of Hungary, Anastasia of Kiev. This sword, a cavalry sabre now in the Kunsthistorisches Museum in Vienna, appears to be the work of Hungarian goldsmiths of the ninth or tenth century. |
Attila | Legends about Attila and his meeting with Pope Leo I | Legends about Attila and his meeting with Pope Leo I
thumb|Meeting of Attila with Pope Leo (Chronicon Pictum, 1358).
An anonymous chronicler of the medieval period represented the meeting of Pope Leo and Atilla as attended also by Saint Peter and Saint Paul, "a miraculous tale calculated to meet the taste of the time" This apotheosis was later portrayed artistically by the Renaissance artist Raphael and sculptor Algardi, whom eighteenth-century historian Edward Gibbon praised for establishing "one of the noblest legends of ecclesiastical tradition".
According to a version of this narrative related in the Chronicon Pictum, a mediaeval Hungarian chronicle, the Pope promised Attila that if he left Rome in peace, one of his successors would receive a holy crown (which has been understood as referring to the Holy Crown of Hungary). |
Attila | Attila in Germanic heroic legend | Attila in Germanic heroic legend
Some histories and chronicles describe Attila as a great and noble king, and he plays major roles in three Norse texts: Atlakviða, Volsunga saga, and Atlamál. The Polish Chronicle represents Attila's name as Aquila.
Frutolf of Michelsberg and Otto of Freising pointed out that some songs as "vulgar fables" and made Theoderic the Great, Attila and Ermanaric contemporaries, when any reader of Jordanes knew that this was not the case. This refers to the so-called historical poems about Dietrich von Bern (Theoderic), in which Etzel (German for Attila) is Dietrich's refuge in exile from his wicked uncle Ermenrich (Ermanaric). Etzel is most prominent in the poems Dietrichs Flucht and the Rabenschlacht. Etzel also appears as Kriemhild's second noble husband in the Nibelungenlied, in which Kriemhild causes the destruction of both the Hunnish kingdom and that of her Burgundian relatives. |
Attila | Early modern and modern reception | Early modern and modern reception
thumb|A painting of Attila riding a pale horse, by French Romantic artist Eugène Delacroix (1798–1863).
In 1812, Ludwig van Beethoven conceived the idea of writing an opera about Attila and approached August von Kotzebue to write the libretto. It was, however, never written. In 1846, Giuseppe Verdi wrote the opera, loosely based on episodes in Attila's invasion of Italy.
In World War I, Allied propaganda referred to Germans as the "Huns", based on a 1900 speech by Emperor Wilhelm II praising Attila the Hun's military prowess, according to Jawaharlal Nehru's Glimpses of World History. Der Spiegel commented on 6 November 1948, that the Sword of Attila was hanging menacingly over Austria.
American writer Cecelia Holland wrote The Death of Attila (1973), a historical novel in which Attila appears as a powerful background figure whose life and death deeply affect the protagonists, a young Hunnic warrior and a Germanic one.
In modern Hungary and in Turkey, "Attila" and its Turkish variation "Atilla" are commonly used as a male first name. In Hungary, several public places are named after Attila; for instance, in Budapest there are 10 Attila Streets, one of which is an important street behind the Buda Castle. When the Turkish Armed Forces invaded Cyprus in 1974, the operations were named after Attila ("The Attila Plan").
The 1954 Universal International film Sign of the Pagan starred Jack Palance as Attila. |
Attila | See also | See also
Onegesius
Bleda
Mundzuk |
Attila | Notes | Notes |
Attila | Sources | Sources
|
Attila | External links | External links
Works about Attila at Project Gutenberg
Category:5th-century Hunnic kings
Category:5th-century monarchs in Europe
Category:400s births
Category:453 deaths
Category:Deaths from choking
Category:Attilid dynasty
Category:Characters in the Divine Comedy |
Attila | Table of Content | redirect2, Etymology, Historiography and sources, Appearance and character, Early life and background, Campaigns against the Eastern Roman Empire, Solitary kingship, In the west, Invasion of Italy and death, Death, Descendants, Later folklore and iconography, Attila and Hun tradition in the medieval Hungarian Royal Court, Legends about Attila and the sword of Mars, Legends about Attila and his meeting with Pope Leo I, Attila in Germanic heroic legend, Early modern and modern reception, See also, Notes, Sources, External links |
Aegean Sea | Short description | thumb|The extent of the Aegean Sea on a map of the Mediterranean Sea
The Aegean Sea is an elongated embayment of the Mediterranean Sea between Europe and Asia. It is located between the Balkans and Anatolia, and covers an area of some . In the north, the Aegean is connected to the Marmara Sea, which in turn connects to the Black Sea, by the straits of the Dardanelles and the Bosphorus, respectively. The Aegean Islands are located within the sea and some bound it on its southern periphery, including Crete and Rhodes. The sea reaches a maximum depth of 2,639 m (8,658 ft) to the west of Karpathos. The Thracian Sea and the Sea of Crete are main subdivisions of the Aegean Sea.
The Aegean Islands can be divided into several island groups, including the Dodecanese, the Cyclades, the Sporades, the Saronic islands and the North Aegean Islands, as well as Crete and its surrounding islands. The Dodecanese, located to the southeast, includes the islands of Rhodes, Kos, and Patmos; the islands of Delos and Naxos are within the Cyclades to the south of the sea. Lesbos is part of the North Aegean Islands. Euboea, the second-largest island in Greece, is located in the Aegean, despite being administered as part of Central Greece. Nine out of twelve of the Administrative regions of Greece border the sea, along with the Turkish provinces of Edirne, Çanakkale, Balıkesir, İzmir, Aydın and Muğla to the east of the sea. Various Turkish islands in the sea are Imbros, Tenedos, Cunda Island, and the Foça Islands.
The Aegean Sea has been historically important, especially regarding the civilization of Ancient Greece, which inhabited the area around the coast of the Aegean and the Aegean islands. The Aegean islands facilitated contact between the people of the area and between Europe and Asia. Along with the Greeks, Thracians lived along the northern coasts. The Romans conquered the area under the Roman Empire, and later the Byzantine Empire held it against advances by the First Bulgarian Empire. The Fourth Crusade weakened Byzantine control of the area, and it was eventually conquered by the Ottoman Empire, with the exception of Crete, which was a Venetian colony until 1669. The Greek War of Independence allowed a Greek state on the coast of the Aegean from 1829 onwards. The Ottoman Empire held a presence over the sea for over 500 years until it was replaced by modern Turkey.
The rocks making up the floor of the Aegean are mainly limestone, though often greatly altered by volcanic activity that has convulsed the region in relatively recent geologic times. Of particular interest are the richly colored sediments in the region of the islands of Santorini and Milos, in the south Aegean. Notable cities on the Aegean coastline include Athens, Thessaloniki, Volos, Kavala, and Heraklion in Greece, and İzmir and Bodrum in Turkey.
Several issues concerning sovereignty within the Aegean Sea are disputed between Greece and Turkey. The Aegean dispute has had a large effect on Greece-Turkey relations since the 1970s. Issues include the delimitation of territorial waters, national airspace, exclusive economic zones, and flight information regions. |
Aegean Sea | Name and etymology | Name and etymology
The name Aegaeus, used by Late Latin authors, referred to Aegeus, who was said to have jumped into that sea to drown himself (rather than throw himself from the Athenian acropolis, as told by some Greek authors). He was the father of Theseus, the mythical king and founder-hero of Athens. Aegeus had told Theseus to put up white sails when returning if he was successful in killing the Minotaur. When Theseus returned, he forgot these instructions, and Aegeus thought his son had died, so he drowned himself in the sea.Hyginus, Fab. 43; Serv. Verg. A. 3.74; Scriptores rerum mythicarum Latini, ed. Bode, i. p. 117 (Second Vatican Mythographer 125).
The sea was known in Latin as Mare Aegaeum while under the control of the Roman Empire. The Venetians, who ruled many Greek islands in the High and Late Middle Ages, popularized the name Archipelago (, meaning "main sea" or "chief sea"), a name that held on in many European countries until the early modern period. In South Slavic languages, the Aegean is called White Sea (; ; ).Zbornik Matice srpske za društvene nauke: (1961), Volumes 28–31, p.74 The Turkish name for the sea is Ege Denizi, which is derived from the Greek name, and Adalar Denizi meaning "Sea of Islands". |
Aegean Sea | Geography | Geography
The Aegean Sea is an elongated embayment of the Mediterranean Sea and covers about in area, measuring about longitudinally and latitudinal. The sea's maximum depth is , located at a point west of Karpathos. The Aegean Islands are found within its waters, with the following islands delimiting the sea on the south, generally from west to east: Kythera, Antikythera, Crete, Kasos, Karpathos and Rhodes. The Anatolian peninsula marks the eastern boundary of the sea, while the Greek mainland marks the west. Several seas are contained within the Aegean Sea; the Thracian Sea is a section of the Aegean located to the north, the Icarian Sea to the east, the Myrtoan Sea to the west, while the Sea of Crete is the southern section.
The Greek regions that border the sea, in alphabetical order, are Attica, Central Greece, Central Macedonia, Crete, Eastern Macedonia and Thrace, North Aegean, Peloponnese, South Aegean, and Thessaly. The traditional Greek region of Macedonia also borders the sea, to the north.
The Aegean Islands, which almost all belong to Greece, can be divided into seven groups:
Northeastern Aegean Islands, which lie in the Thracian Sea
East Aegean Islands (Euboea)
Northern Sporades
Cyclades
Saronic Islands (or Argo-Saronic Islands)
Dodecanese (or Southern Sporades)Administratively, the Greek Dodecanese also contains Kastellorizo, situated further east outside the Aegean proper.
Crete
Many of the Aegean islands or island chains, are geographical extensions of the mountains on the mainland. One chain extends across the sea to Chios, another extends across Euboea to Samos, and a third extends across the Peloponnese and Crete to Rhodes, dividing the Aegean from the Mediterranean.
The bays and gulfs of the Aegean beginning at the South and moving clockwise include on Crete, the Mirabello, Almyros, Souda and Chania bays or gulfs, on the mainland the Myrtoan Sea to the west with the Argolic Gulf, the Saronic Gulf northwestward, the Petalies Gulf which connects with the South Euboic Sea, the Pagasetic Gulf which connects with the North Euboic Sea, the Thermian Gulf northwestward, the Chalkidiki Peninsula including the Cassandra and the Singitic Gulfs, northward the Strymonian Gulf and the Gulf of Kavala and the rest are in Turkey; Saros Gulf, Edremit Gulf, Dikili Gulf, Gulf of Çandarlı, Gulf of İzmir, Gulf of Kuşadası, Gulf of Gökova, Güllük Gulf.
The Aegean Sea is connected to the Sea of Marmara by the Dardanelles, also known from Classical Antiquity as the Hellespont. The Dardanelles are located to the northeast of the sea. It ultimately connects with the Black Sea through the Bosporus strait, upon which lies the city of Istanbul. The Dardanelles and the Bosporus are known as the Turkish Straits. |
Aegean Sea | Extent | Extent
According to the International Hydrographic Organization, the limits of the Aegean Sea as follows:
On the south: A line running from Cape Aspro (28°16′E) in Asia Minor, to Cum Burnù (Capo della Sabbia) the Northeast extreme of the Island of Rhodes, through the island to Cape Prasonisi, the Southwest point thereof, on to Vrontos Point (35°33′N) in Skarpanto [Karpathos], through this island to Castello Point, the South extreme thereof, across to Cape Plaka (East extremity of Crete), through Crete to Agria Grabusa, the Northwest extreme thereof, thence to Cape Apolytares in Antikythera Island, through the island to Psira Rock (off the Northwest point) and across to Cape Trakhili in Kythira Island, through Kythira to the Northwest point (Cape Karavugia) and thence to Cape Santa Maria () in the Morea.
In the Dardanelles: A line joining Kum Kale (26°11′E) and Cape Helles. |
Aegean Sea | Hydrography | Hydrography
Aegean surface water circulates in a counterclockwise gyre, with hypersaline Mediterranean water moving northward along the west coast of Turkey, before being displaced by less dense Black Sea outflow. The dense Mediterranean water sinks below the Black Sea inflow to a depth of , then flows through the Dardanelles Strait and into the Sea of Marmara at velocities of . The Black Sea outflow moves westward along the northern Aegean Sea, then flows southwards along the east coast of Greece.
The physical oceanography of the Aegean Sea is controlled mainly by the regional climate, the fresh water discharge from major rivers draining southeastern Europe, and the seasonal variations in the Black Sea surface water outflow through the Dardanelles Strait.
AnalysisYagar, D., 1994. Late glacial-Holocene evolution of the Aegean Sea. Ph.D. Thesis, Inst. Mar. Sci. Technol., Dokuz Eyltil Univ., 329 pp. (Unpubl.) of the Aegean during 1991 and 1992 revealed three distinct water masses:
Aegean Sea Surface Water – thick veneer, with summer temperatures of 21–26 °C and winter temperatures ranging from in the north to in the south.
Aegean Sea Intermediate Water – Aegean Sea Intermediate Water extends from to with temperatures ranging from .
Aegean Sea Bottom Water – occurring at depths below with a very uniform temperature () and salinity (3.91–3.92%). |
Aegean Sea | Climate | Climate
thumb|Climate map of Greece. Most of the landmass surrounding the Aegean Sea is classified as Csa, with the northern region being BSk.
The climate of the Aegean Sea largely reflects the climate of Greece and Western Turkey, which is to say, predominantly Mediterranean. According to the Köppen climate classification, most of the Aegean is classified as Hot-summer Mediterranean (Csa), with hotter and drier summers along with milder and wetter winters. However, high temperatures during summers are generally not quite as high as those in arid or semiarid climates due to the presence of a large body of water. This is most predominant in the west and east coasts of the Aegean, and within the Aegean islands. In the north of the Aegean Sea, the climate is instead classified as Cold semi-arid (BSk), which feature cooler summers than Hot-summer Mediterranean climates. The Etesian winds are a dominant weather influence in the Aegean Basin.
The below table lists climate conditions of some major Aegean cities:
+Climate characteristics of some major cities on the Aegean coastCityMean temperature (daily high)Mean total rainfallJanuaryJulyJanuaryJuly°C°F°C°FmmindaysmmindaysAlexandroupolis8.447.130.186.260.42.386.817.60.692.5Bodrum15.159.234.293.6134.15.2812.31.30.051.5Heraklion15.259.428.683.591.53.610.11.00.040.1İzmir12.454.333.291.8132.75.2212.61.70.070.4Thessaloniki9.348.732.590.535.21.398.827.31.073.8Source: World Meteorological Organization, Turkish State Meteorological Service"Resmi İstatistikler: İllerimize Ait Genel İstatistik Verileri" (in Turkish). Turkish State Meteorological Service. Retrieved 4 May 2019. |
Aegean Sea | Population | Population
Numerous Greek and Turkish settlements are located along their mainland coast, as well as on towns on the Aegean islands. The largest cities are Athens and Thessaloniki in Greece and İzmir in Turkey. The most populated of the Aegean islands is Crete, followed by Euboea and Rhodes.
thumb|right|150x150px|İzmir
thumb|right|150x150px|Athens
thumb|right|150x150px|Thessaloniki
thumb|right|150x150px|Bodrum
+Most populous urban areas on the Aegean coastRankCityCountryRegion/CountyPopulation (urban)1AthensGreeceCentral Greece3,090,5082İzmirTurkeyİzmir Province2,948,6093ThessalonikiGreeceMacedonia824,6764BodrumTurkeyMuğla Province198,3355ÇanakkaleTurkeyÇanakkale Province182,3896HeraklionGreeceCrete173,9937VolosGreeceThessaly144,4498KuşadasıTurkeyAydın Province133,1779ChaniaGreeceCrete108,64210DidimTurkeyAydın Province100,189 |
Aegean Sea | Biogeography and ecology | Biogeography and ecology |
Aegean Sea | Protected areas | Protected areas
Greece has established several marine protected areas along its coasts. According to the Network of Managers of Marine Protected Areas in the Mediterranean (MedPAN), four Greek MPAs are participating in the Network. These include Alonnisos Marine Park, while the Missolonghi–Aitoliko Lagoons and the island of Zakynthos are not on the Aegean. |
Aegean Sea | History | History |
Aegean Sea | Ancient history | Ancient history
200px|thumb|upright=1.25|Female figure from Naxos (2800–2300 BC)
The current coastline dates back to about 4000 BC. Before that time, at the peak of the last ice age (about 18,000 years ago) sea levels everywhere were lower, and there were large well-watered coastal plains instead of much of the northern Aegean. When they were first occupied, the present-day islands including Milos with its important obsidian production were probably still connected to the mainland. The present coastal arrangement appeared around 9,000 years ago, with post-ice age sea levels continuing to rise for another 3,000 years after that.
The subsequent Bronze Age civilizations of Greece and the Aegean Sea have given rise to the general term Aegean civilization. In ancient times, the sea was the birthplace of two ancient civilizations – the Minoans of Crete and the Mycenaeans of the Peloponnese.Tracey Cullen, Aegean Prehistory: A Review (American Journal of Archaeology. Supplement, 1); Oliver Dickinson, The Aegean Bronze Age (Cambridge World Archaeology).
The Minoan civilization was a Bronze Age civilization on the island of Crete and other Aegean islands, flourishing from around 3000 to 1450 BC before a period of decline, finally ending at around 1100 BC. It represented the first advanced civilization in Europe, leaving behind massive building complexes, tools, stunning artwork, writing systems, and a massive network of trade. The Minoan period saw extensive trade between Crete, Aegean, and Mediterranean settlements, particularly the Near East. The most notable Minoan palace is that of Knossos, followed by that of Phaistos. The Mycenaean Greeks arose on the mainland, becoming the first advanced civilization in mainland Greece, which lasted from approximately 1600 to 1100 BC. It is believed that the site of Mycenae, which sits close to the Aegean coast, was the center of Mycenaean civilization. The Mycenaeans introduced several innovations in the fields of engineering, architecture and military infrastructure, while trade over vast areas of the Mediterranean, including the Aegean, was essential for the Mycenaean economy. Their syllabic script, the Linear B, offers the first written records of the Greek language and their religion already included several deities that can also be found in the Olympic Pantheon. Mycenaean Greece was dominated by a warrior elite society and consisted of a network of palace-centered states that developed rigid hierarchical, political, social and economic systems. At the head of this society was the king, known as wanax.
The civilization of Mycenaean Greeks perished with the collapse of Bronze Age culture in the eastern Mediterranean, to be followed by the so-called Greek Dark Ages. It is undetermined what cause the collapse of the Mycenaeans. During the Greek Dark Ages, writing in the Linear B script ceased, vital trade links were lost, and towns and villages were abandoned. |
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