hpprc/enwiki-paragraphs · Datasets at Hugging Face

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A. A. Milne	924	Religious views	He wrote in the poem "Explained":
A. A. Milne	924	Religious views	Elizabeth Ann Said to her Nan: "Please will you tell me how God began? Somebody must have made Him. So Who could it be, 'cos I want to know?"
A. A. Milne	924	Religious views	He also wrote in the poem "Vespers":
A. A. Milne	924	Religious views	"Oh! Thank you, God, for a lovely day. And what was the other I had to say? I said "Bless Daddy," so what can it be? Oh! Now I remember it. God bless Me."
Asociación Alumni	925		Asociación Alumni, usually just Alumni, is an Argentine rugby union club located in Tortuguitas, Greater Buenos Aires. The senior squad currently competes at Top 12, the first division of the Unión de Rugby de Buenos Aires league system.
Asociación Alumni	925		The club has ties with former football club Alumni because both were established by Buenos Aires English High School students.
Asociación Alumni	925	History	The first club with the name "Alumni" played association football, having been found in 1898 by students of Buenos Aires English High School (BAEHS) along with director Alexander Watson Hutton. Originally under the name "English High School A.C.", the team would be later obliged by the Association to change its name, therefore "Alumni" was chosen, following a proposal by Carlos Bowers, a former student of the school.
Asociación Alumni	925	History	Alumni was the most successful team during the first years of Argentine football, winning 10 of 14 league championships contested. Alumni is still considered the first great football team in the country. Alumni was reorganised in 1908, "in order to encourage people to practise all kind of sports, specially football". This was the last try to develop itself as a sports club rather than just a football team, such as Lomas, Belgrano and Quilmes had successfully done in the past, but the efforts were not enough. Alumni played its last game in 1911 and was definitely dissolved on April 24, 1913.
Asociación Alumni	925	History	In 1951, two guards of the BAEHS, Daniel Ginhson (also a former player of Buenos Aires F.C.) and Guillermo Cubelli, supported by the school's alumni and fathers of the students, they decided to establish a club focused on rugby union exclusively. Former players still alive of Alumni football club and descendants of other players already dead gave their permission to use the name "Alumni".
Asociación Alumni	925	History	On December 13, in a meeting presided by Carlos Bowers himself (who had proposed the name "Alumni" to the original football team 50 years before), the club was officially established under the name "Asociación Juvenil Alumni", also adopting the same colors as its predecessor.
Asociación Alumni	925	History	The team achieved good results and in 1960 the club presented a team that won the third division of the Buenos Aires league, reaching the second division. Since then, Alumni has played at the highest level of Argentine rugby and its rivalry with Belgrano Athletic Club is one of the fiercest local derbies in Buenos Aires. Alumni would later climb up to first division winning 5 titles: 4 consecutive between 1989 and 1992, and the other in 2001.
Asociación Alumni	925	History	In 2002, Alumni won its first Nacional de Clubes title, defeating Jockey Club de Rosario 23–21 in the final.
Asociación Alumni	925	Players	As of January 2018:
Axiom	928		An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.
Axiom	928		The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning.
Axiom	928		In mathematics, an axiom may be a "logical axiom" or a "non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are substantive assertions about the elements of the domain of a specific mathematical theory, such as arithmetic.
Axiom	928		Non-logical axioms may also be called "postulates" or "assumptions". In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., the parallel postulate in Euclidean geometry). To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms), and there are typically many ways to axiomatize a given mathematical domain.
Axiom	928		Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the philosophy of mathematics.
Axiom	928	Etymology	The word axiom comes from the Greek word ἀξίωμα (axíōma), a verbal noun from the verb ἀξιόειν (axioein), meaning "to deem worthy", but also "to require", which in turn comes from ἄξιος (áxios), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the ancient Greek philosophers and mathematicians, axioms were taken to be immediately evident propositions, foundational and common to many fields of investigation, and self-evidently true without any further argument or proof.
Axiom	928	Etymology	The root meaning of the word postulate is to "demand"; for instance, Euclid demands that one agree that some things can be done (e.g., any two points can be joined by a straight line).
Axiom	928	Etymology	Ancient geometers maintained some distinction between axioms and postulates. While commenting on Euclid's books, Proclus remarks that "Geminus held that this [4th] Postulate should not be classed as a postulate but as an axiom, since it does not, like the first three Postulates, assert the possibility of some construction but expresses an essential property." Boethius translated 'postulate' as petitio and called the axioms notiones communes but in later manuscripts this usage was not always strictly kept.
Axiom	928	Historical development	The logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments (syllogisms, rules of inference) was developed by the ancient Greeks, and has become the core principle of modern mathematics. Tautologies excluded, nothing can be deduced if nothing is assumed. Axioms and postulates are thus the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions (theorems, in the case of mathematics) must be proven with the aid of these basic assumptions. However, the interpretation of mathematical knowledge has changed from ancient times to the modern, and consequently the terms axiom and postulate hold a slightly different meaning for the present day mathematician, than they did for Aristotle and Euclid.
Axiom	928	Historical development	The ancient Greeks considered geometry as just one of several sciences, and held the theorems of geometry on par with scientific facts. As such, they developed and used the logico-deductive method as a means of avoiding error, and for structuring and communicating knowledge. Aristotle's posterior analytics is a definitive exposition of the classical view.
Axiom	928	Historical development	An "axiom", in classical terminology, referred to a self-evident assumption common to many branches of science. A good example would be the assertion that:
Axiom	928	Historical development	When an equal amount is taken from equals, an equal amount results.
Axiom	928	Historical development	At the foundation of the various sciences lay certain additional hypotheses that were accepted without proof. Such a hypothesis was termed a postulate. While the axioms were common to many sciences, the postulates of each particular science were different. Their validity had to be established by means of real-world experience. Aristotle warns that the content of a science cannot be successfully communicated if the learner is in doubt about the truth of the postulates.
Axiom	928	Historical development	The classical approach is well-illustrated by Euclid's Elements, where a list of postulates is given (common-sensical geometric facts drawn from our experience), followed by a list of "common notions" (very basic, self-evident assertions).
Axiom	928	Historical development	A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, propositions, theorems) and definitions. One must concede the need for primitive notions, or undefined terms or concepts, in any study. Such abstraction or formalization makes mathematical knowledge more general, capable of multiple different meanings, and therefore useful in multiple contexts. Alessandro Padoa, Mario Pieri, and Giuseppe Peano were pioneers in this movement.
Axiom	928	Historical development	Structuralist mathematics goes further, and develops theories and axioms (e.g. field theory, group theory, topology, vector spaces) without any particular application in mind. The distinction between an "axiom" and a "postulate" disappears. The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts rests on the acceptance of the basic hypotheses. However, by throwing out Euclid's fifth postulate, one can get theories that have meaning in wider contexts (e.g., hyperbolic geometry). As such, one must simply be prepared to use labels such as "line" and "parallel" with greater flexibility. The development of hyperbolic geometry taught mathematicians that it is useful to regard postulates as purely formal statements, and not as facts based on experience.
Axiom	928	Historical development	When mathematicians employ the field axioms, the intentions are even more abstract. The propositions of field theory do not concern any one particular application; the mathematician now works in complete abstraction. There are many examples of fields; field theory gives correct knowledge about them all.
Axiom	928	Historical development	It is not correct to say that the axioms of field theory are "propositions that are regarded as true without proof." Rather, the field axioms are a set of constraints. If any given system of addition and multiplication satisfies these constraints, then one is in a position to instantly know a great deal of extra information about this system.
Axiom	928	Historical development	Modern mathematics formalizes its foundations to such an extent that mathematical theories can be regarded as mathematical objects, and mathematics itself can be regarded as a branch of logic. Frege, Russell, Poincaré, Hilbert, and Gödel are some of the key figures in this development.
Axiom	928	Historical development	Another lesson learned in modern mathematics is to examine purported proofs carefully for hidden assumptions.
Axiom	928	Historical development	In the modern understanding, a set of axioms is any collection of formally stated assertions from which other formally stated assertions follow – by the application of certain well-defined rules. In this view, logic becomes just another formal system. A set of axioms should be consistent; it should be impossible to derive a contradiction from the axioms. A set of axioms should also be non-redundant; an assertion that can be deduced from other axioms need not be regarded as an axiom.
Axiom	928	Historical development	It was the early hope of modern logicians that various branches of mathematics, perhaps all of mathematics, could be derived from a consistent collection of basic axioms. An early success of the formalist program was Hilbert's formalization of Euclidean geometry, and the related demonstration of the consistency of those axioms.
Axiom	928	Historical development	In a wider context, there was an attempt to base all of mathematics on Cantor's set theory. Here, the emergence of Russell's paradox and similar antinomies of naïve set theory raised the possibility that any such system could turn out to be inconsistent.
Axiom	928	Historical development	The formalist project suffered a decisive setback, when in 1931 Gödel showed that it is possible, for any sufficiently large set of axioms (Peano's axioms, for example) to construct a statement whose truth is independent of that set of axioms. As a corollary, Gödel proved that the consistency of a theory like Peano arithmetic is an unprovable assertion within the scope of that theory.
Axiom	928	Historical development	It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by the system of natural numbers, an infinite but intuitively accessible formal system. However, at present, there is no known way of demonstrating the consistency of the modern Zermelo–Fraenkel axioms for set theory. Furthermore, using techniques of forcing (Cohen) one can show that the continuum hypothesis (Cantor) is independent of the Zermelo–Fraenkel axioms. Thus, even this very general set of axioms cannot be regarded as the definitive foundation for mathematics.
Axiom	928	Historical development	Experimental sciences - as opposed to mathematics and logic - also have general founding assertions from which a deductive reasoning can be built so as to express propositions that predict properties - either still general or much more specialized to a specific experimental context. For instance, Newton's laws in classical mechanics, Maxwell's equations in classical electromagnetism, Einstein's equation in general relativity, Mendel's laws of genetics, Darwin's Natural selection law, etc. These founding assertions are usually called principles or postulates so as to distinguish from mathematical axioms.
Axiom	928	Historical development	As a matter of facts, the role of axioms in mathematics and postulates in experimental sciences is different. In mathematics one neither "proves" nor "disproves" an axiom. A set of mathematical axioms gives a set of rules that fix a conceptual realm, in which the theorems logically follow. In contrast, in experimental sciences, a set of postulates shall allow deducing results that match or do not match experimental results. If postulates do not allow deducing experimental predictions, they do not set a scientific conceptual framework and have to be completed or made more accurate. If the postulates allow deducing predictions of experimental results, the comparison with experiments allows falsifying (falsified) the theory that the postulates install. A theory is considered valid as long as it has not been falsified.
Axiom	928	Historical development	Now, the transition between the mathematical axioms and scientific postulates is always slightly blurred, especially in physics. This is due to the heavy use of mathematical tools to support the physical theories. For instance, the introduction of Newton's laws rarely establishes as a prerequisite neither Euclidean geometry or differential calculus that they imply. It became more apparent when Albert Einstein first introduced special relativity where the invariant quantity is no more the Euclidean length l {\displaystyle l} (defined as l 2 = x 2 + y 2 + z 2 {\displaystyle l^{2}=x^{2}+y^{2}+z^{2}} ) > but the Minkowski spacetime interval s {\displaystyle s} (defined as s 2 = c 2 t 2 − x 2 − y 2 − z 2 {\displaystyle s^{2}=c^{2}t^{2}-x^{2}-y^{2}-z^{2}} ), and then general relativity where flat Minkowskian geometry is replaced with pseudo-Riemannian geometry on curved manifolds.
Axiom	928	Historical development	In quantum physics, two sets of postulates have coexisted for some time, which provide a very nice example of falsification. The 'Copenhagen school' (Niels Bohr, Werner Heisenberg, Max Born) developed an operational approach with a complete mathematical formalism that involves the description of quantum system by vectors ('states') in a separable Hilbert space, and physical quantities as linear operators that act in this Hilbert space. This approach is fully falsifiable and has so far produced the most accurate predictions in physics. But it has the unsatisfactory aspect of not allowing answers to questions one would naturally ask. For this reason, another 'hidden variables' approach was developed for some time by Albert Einstein, Erwin Schrödinger, David Bohm. It was created so as to try to give deterministic explanation to phenomena such as entanglement. This approach assumed that the Copenhagen school description was not complete, and postulated that some yet unknown variable was to be added to the theory so as to allow answering some of the questions it does not answer (the founding elements of which were discussed as the EPR paradox in 1935). Taking this ideas seriously, John Bell derived in 1964 a prediction that would lead to different experimental results (Bell's inequalities) in the Copenhagen and the Hidden variable case. The experiment was conducted first by Alain Aspect in the early 1980's, and the result excluded the simple hidden variable approach (sophisticated hidden variables could still exist but their properties would still be more disturbing than the problems they try to solve). This does not mean that the conceptual framework of quantum physics can be considered as complete now, since some open questions still exist (the limit between the quantum and classical realms, what happens during a quantum measurement, what happens in a completely closed quantum system such as the universe itself, etc.).
Axiom	928	Mathematical logic	In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical and non-logical (somewhat similar to the ancient distinction between "axioms" and "postulates" respectively).
Axiom	928	Mathematical logic	These are certain formulas in a formal language that are universally valid, that is, formulas that are satisfied by every assignment of values. Usually one takes as logical axioms at least some minimal set of tautologies that is sufficient for proving all tautologies in the language; in the case of predicate logic more logical axioms than that are required, in order to prove logical truths that are not tautologies in the strict sense.
Axiom	928	Mathematical logic	In propositional logic it is common to take as logical axioms all formulae of the following forms, where ϕ {\displaystyle \phi } , χ {\displaystyle \chi } , and ψ {\displaystyle \psi } can be any formulae of the language and where the included primitive connectives are only " ¬ {\displaystyle \neg } " for negation of the immediately following proposition and " → {\displaystyle \to } " for a preconceptions=sequence of forthcoming events of times of moment taking place. implication from antecedent to consequent propositions:
Axiom	928	Mathematical logic	Each of these patterns is an axiom schema, a rule for generating an infinite number of axioms. For example, if A {\displaystyle A} , B {\displaystyle B} , and C {\displaystyle C} are propositional variables, then A → ( B → A ) {\displaystyle A\to (B\to A)} and ( A → ¬ B ) → ( C → ( A → ¬ B ) ) {\displaystyle (A\to \lnot B)\to (C\to (A\to \lnot B))} are both instances of axiom schema 1, and hence are axioms. It can be shown that with only these three axiom schemata and modus ponens, one can prove all tautologies of the propositional calculus. It can also be shown that no pair of these schemata is sufficient for proving all tautologies with modus ponens.
Axiom	928	Mathematical logic	Other axiom schemata involving the same or different sets of primitive connectives can be alternatively constructed.
Axiom	928	Mathematical logic	These axiom schemata are also used in the predicate calculus, but additional logical axioms are needed to include a quantifier in the calculus.
Axiom	928	Mathematical logic	Axiom of Equality.Let L {\displaystyle {\mathfrak {L}}} be a first-order language. For each variable x {\displaystyle x} , the below formula is universally valid.
Axiom	928	Mathematical logic	x = x {\displaystyle x=x}
Axiom	928	Mathematical logic	This means that, for any variable symbol x {\displaystyle x} , the formula x = x {\displaystyle x=x} can be regarded as an axiom. Also, in this example, for this not to fall into vagueness and a never-ending series of "primitive notions", either a precise notion of what we mean by x = x {\displaystyle x=x} (or, for that matter, "to be equal") has to be well established first, or a purely formal and syntactical usage of the symbol = {\displaystyle =} has to be enforced, only regarding it as a string and only a string of symbols, and mathematical logic does indeed do that.
Axiom	928	Mathematical logic	Another, more interesting example axiom scheme, is that which provides us with what is known as Universal Instantiation:
Axiom	928	Mathematical logic	Axiom scheme for Universal Instantiation.Given a formula ϕ {\displaystyle \phi } in a first-order language L {\displaystyle {\mathfrak {L}}} , a variable x {\displaystyle x} and a term t {\displaystyle t} that is substitutable for x {\displaystyle x} in ϕ {\displaystyle \phi } , the below formula is universally valid.
Axiom	928	Mathematical logic	∀ x ϕ → ϕ t x {\displaystyle \forall x\,\phi \to \phi _{t}^{x}}
Axiom	928	Mathematical logic	Where the symbol ϕ t x {\displaystyle \phi _{t}^{x}} stands for the formula ϕ {\displaystyle \phi } with the term t {\displaystyle t} substituted for x {\displaystyle x} . (See Substitution of variables.) In informal terms, this example allows us to state that, if we know that a certain property P {\displaystyle P} holds for every x {\displaystyle x} and that t {\displaystyle t} stands for a particular object in our structure, then we should be able to claim P ( t ) {\displaystyle P(t)} . Again, we are claiming that the formula ∀ x ϕ → ϕ t x {\displaystyle \forall x\phi \to \phi _{t}^{x}} is valid, that is, we must be able to give a "proof" of this fact, or more properly speaking, a metaproof. These examples are metatheorems of our theory of mathematical logic since we are dealing with the very concept of proof itself. Aside from this, we can also have Existential Generalization:
Axiom	928	Mathematical logic	Axiom scheme for Existential Generalization. Given a formula ϕ {\displaystyle \phi } in a first-order language L {\displaystyle {\mathfrak {L}}} , a variable x {\displaystyle x} and a term t {\displaystyle t} that is substitutable for x {\displaystyle x} in ϕ {\displaystyle \phi } , the below formula is universally valid.
Axiom	928	Mathematical logic	ϕ t x → ∃ x ϕ {\displaystyle \phi _{t}^{x}\to \exists x\,\phi }
Axiom	928	Mathematical logic	Non-logical axioms are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example, the natural numbers and the integers, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as groups). Thus non-logical axioms, unlike logical axioms, are not tautologies. Another name for a non-logical axiom is postulate.
Axiom	928	Mathematical logic	Almost every modern mathematical theory starts from a given set of non-logical axioms, and it was thought that, in principle, every theory could be axiomatized in this way and formalized down to the bare language of logical formulas.
Axiom	928	Mathematical logic	Non-logical axioms are often simply referred to as axioms in mathematical discourse. This does not mean that it is claimed that they are true in some absolute sense. For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom, we can do quite well developing (the more general) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.
Axiom	928	Mathematical logic	Thus, an axiom is an elementary basis for a formal logic system that together with the rules of inference define a deductive system.
Axiom	928	Mathematical logic	This section gives examples of mathematical theories that are developed entirely from a set of non-logical axioms (axioms, henceforth). A rigorous treatment of any of these topics begins with a specification of these axioms.
Axiom	928	Mathematical logic	Basic theories, such as arithmetic, real analysis and complex analysis are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of Zermelo–Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann–Bernays–Gödel set theory, a conservative extension of ZFC. Sometimes slightly stronger theories such as Morse–Kelley set theory or set theory with a strongly inaccessible cardinal allowing the use of a Grothendieck universe is used, but in fact, most mathematicians can actually prove all they need in systems weaker than ZFC, such as second-order arithmetic.
Axiom	928	Mathematical logic	The study of topology in mathematics extends all over through point set topology, algebraic topology, differential topology, and all the related paraphernalia, such as homology theory, homotopy theory. The development of abstract algebra brought with itself group theory, rings, fields, and Galois theory.
Axiom	928	Mathematical logic	This list could be expanded to include most fields of mathematics, including measure theory, ergodic theory, probability, representation theory, and differential geometry.
Axiom	928	Mathematical logic	The Peano axioms are the most widely used axiomatization of first-order arithmetic. They are a set of axioms strong enough to prove many important facts about number theory and they allowed Gödel to establish his famous second incompleteness theorem.
Axiom	928	Mathematical logic	We have a language L N T = { 0 , S } {\displaystyle {\mathfrak {L}}_{NT}=\{0,S\}} where 0 {\displaystyle 0} is a constant symbol and S {\displaystyle S} is a unary function and the following axioms:
Axiom	928	Mathematical logic	The standard structure is N = ⟨ N , 0 , S ⟩ {\displaystyle {\mathfrak {N}}=\langle \mathbb {N} ,0,S\rangle } where N {\displaystyle \mathbb {N} } is the set of natural numbers, S {\displaystyle S} is the successor function and 0 {\displaystyle 0} is naturally interpreted as the number 0.
Axiom	928	Mathematical logic	Probably the oldest, and most famous, list of axioms are the 4 + 1 Euclid's postulates of plane geometry. The axioms are referred to as "4 + 1" because for nearly two millennia the fifth (parallel) postulate ("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four. Ultimately, the fifth postulate was found to be independent of the first four. One can assume that exactly one parallel through a point outside a line exists, or that infinitely many exist. This choice gives us two alternative forms of geometry in which the interior angles of a triangle add up to exactly 180 degrees or less, respectively, and are known as Euclidean and hyperbolic geometries. If one also removes the second postulate ("a line can be extended indefinitely") then elliptic geometry arises, where there is no parallel through a point outside a line, and in which the interior angles of a triangle add up to more than 180 degrees.
Axiom	928	Mathematical logic	The objectives of the study are within the domain of real numbers. The real numbers are uniquely picked out (up to isomorphism) by the properties of a Dedekind complete ordered field, meaning that any nonempty set of real numbers with an upper bound has a least upper bound. However, expressing these properties as axioms requires the use of second-order logic. The Löwenheim–Skolem theorems tell us that if we restrict ourselves to first-order logic, any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger. Some of the latter are studied in non-standard analysis.
Axiom	928	Mathematical logic	A deductive system consists of a set Λ {\displaystyle \Lambda } of logical axioms, a set Σ {\displaystyle \Sigma } of non-logical axioms, and a set { ( Γ , ϕ ) } {\displaystyle \{(\Gamma ,\phi )\}} of rules of inference. A desirable property of a deductive system is that it be complete. A system is said to be complete if, for all formulas ϕ {\displaystyle \phi } ,
Axiom	928	Mathematical logic	if Σ ⊨ ϕ then Σ ⊢ ϕ {\displaystyle {\text{if }}\Sigma \models \phi {\text{ then }}\Sigma \vdash \phi }
Axiom	928	Mathematical logic	that is, for any statement that is a logical consequence of Σ {\displaystyle \Sigma } there actually exists a deduction of the statement from Σ {\displaystyle \Sigma } . This is sometimes expressed as "everything that is true is provable", but it must be understood that "true" here means "made true by the set of axioms", and not, for example, "true in the intended interpretation". Gödel's completeness theorem establishes the completeness of a certain commonly used type of deductive system.
Axiom	928	Mathematical logic	Note that "completeness" has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms Σ {\displaystyle \Sigma } of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement ϕ {\displaystyle \phi } such that neither ϕ {\displaystyle \phi } nor ¬ ϕ {\displaystyle \lnot \phi } can be proved from the given set of axioms.
Axiom	928	Mathematical logic	There is thus, on the one hand, the notion of completeness of a deductive system and on the other hand that of completeness of a set of non-logical axioms. The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another.
Axiom	928	Mathematical logic	Early mathematicians regarded axiomatic geometry as a model of physical space, and obviously, there could only be one such model. The idea that alternative mathematical systems might exist was very troubling to mathematicians of the 19th century and the developers of systems such as Boolean algebra made elaborate efforts to derive them from traditional arithmetic. Galois showed just before his untimely death that these efforts were largely wasted. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details, and modern algebra was born. In the modern view, axioms may be any set of formulas, as long as they are not known to be inconsistent.
Alpha	929		Alpha /ˈælfə/ (uppercase Α, lowercase α; Ancient Greek: ἄλφα, álpha, or Greek: άλφα, romanized: álfa) is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter aleph , which is the West Semitic word for "ox". Letters that arose from alpha include the Latin letter A and the Cyrillic letter А.
Alpha	929	Uses	In Ancient Greek, alpha was pronounced [a] and could be either phonemically long ([aː]) or short ([a]). Where there is ambiguity, long and short alpha are sometimes written with a macron and breve today: Ᾱᾱ, Ᾰᾰ.
Alpha	929	Uses	In Modern Greek, vowel length has been lost, and all instances of alpha simply represent the open front unrounded vowel IPA: [a].
Alpha	929	Uses	In the polytonic orthography of Greek, alpha, like other vowel letters, can occur with several diacritic marks: any of three accent symbols (ά, ὰ, ᾶ), and either of two breathing marks (ἁ, ἀ), as well as combinations of these. It can also combine with the iota subscript (ᾳ).
Alpha	929	Uses	In the Attic–Ionic dialect of Ancient Greek, long alpha [aː] fronted to [ɛː] (eta). In Ionic, the shift took place in all positions. In Attic, the shift did not take place after epsilon, iota, and rho (ε, ι, ρ; e, i, r). In Doric and Aeolic, long alpha is preserved in all positions.
Alpha	929	Uses	Privative a is the Ancient Greek prefix ἀ- or ἀν- a-, an-, added to words to negate them. It originates from the Proto-Indo-European *n̥- (syllabic nasal) and is cognate with English un-.
Alpha	929	Uses	Copulative a is the Greek prefix ἁ- or ἀ- ha-, a-. It comes from Proto-Indo-European *sm̥.
Alpha	929	Uses	The letter alpha represents various concepts in physics and chemistry, including alpha radiation, angular acceleration, alpha particles, alpha carbon and strength of electromagnetic interaction (as fine-structure constant). Alpha also stands for thermal expansion coefficient of a compound in physical chemistry. It is also commonly used in mathematics in algebraic solutions representing quantities such as angles. Furthermore, in mathematics, the letter alpha is used to denote the area underneath a normal curve in statistics to denote significance level when proving null and alternative hypotheses. In ethology, it is used to name the dominant individual in a group of animals. In aerodynamics, the letter is used as a symbol for the angle of attack of an aircraft and the word "alpha" is used as a synonym for this property. In mathematical logic, α is sometimes used as a placeholder for ordinal numbers.
Alpha	929	Uses	The proportionality operator "∝" (in Unicode: U+221D) is sometimes mistaken for alpha.
Alpha	929	Uses	The uppercase letter alpha is not generally used as a symbol because it tends to be rendered identically to the uppercase Latin A.
Alpha	929	Uses	In the International Phonetic Alphabet, the letter ɑ, which looks similar to the lower-case alpha, represents the open back unrounded vowel.
Alpha	929	History and symbolism	The Phoenician alphabet was adopted for Greek in the early 8th century BC, perhaps in Euboea. The majority of the letters of the Phoenician alphabet were adopted into Greek with much the same sounds as they had had in Phoenician, but ʼāleph, the Phoenician letter representing the glottal stop [ʔ], was adopted as representing the vowel [a]; similarly, hē [h] and ʽayin [ʕ] are Phoenician consonants that became Greek vowels, epsilon [e] and omicron [o], respectively.
Alpha	929	History and symbolism	Plutarch, in Moralia, presents a discussion on why the letter alpha stands first in the alphabet. Ammonius asks Plutarch what he, being a Boeotian, has to say for Cadmus, the Phoenician who reputedly settled in Thebes and introduced the alphabet to Greece, placing alpha first because it is the Phoenician name for ox—which, unlike Hesiod, the Phoenicians considered not the second or third, but the first of all necessities. "Nothing at all," Plutarch replied. He then added that he would rather be assisted by Lamprias, his own grandfather, than by Dionysus' grandfather, i.e. Cadmus. For Lamprias had said that the first articulate sound made is "alpha", because it is very plain and simple—the air coming off the mouth does not require any motion of the tongue—and therefore this is the first sound that children make.
Alpha	929	History and symbolism	According to Plutarch's natural order of attribution of the vowels to the planets, alpha was connected with the Moon.
Alpha	929	History and symbolism	As the first letter of the alphabet, Alpha as a Greek numeral came to represent the number 1. Therefore, Alpha, both as a symbol and term, is used to refer to the "first", or "primary", or "principal" (most significant) occurrence or status of a thing.
Alpha	929	History and symbolism	The New Testament has God declaring himself to be the "Alpha and Omega, the beginning and the end, the first and the last." (Revelation 22:13, KJV, and see also 1:8).
Alpha	929	History and symbolism	Consequently, the term "alpha" has also come to be used to denote "primary" position in social hierarchy, examples being the concept of dominant "alpha" members in groups of animals.
Alpha	929	Computer encodings	For accented Greek characters, see Greek diacritics: Computer encoding.
Alvin Toffler	930		Alvin Eugene Toffler (October 4, 1928 – June 27, 2016) was an American writer, futurist, and businessman known for his works discussing modern technologies, including the digital revolution and the communication revolution, with emphasis on their effects on cultures worldwide. He is regarded as one of the world's outstanding futurists.
Alvin Toffler	930		Toffler was an associate editor of Fortune magazine. In his early works he focused on technology and its impact, which he termed "information overload". In 1970, his first major book about the future, Future Shock, became a worldwide best-seller and has sold over 6 million copies.
Alvin Toffler	930		He and his wife Heidi Toffler, who collaborated with him for most of his writings, moved on to examining the reaction to changes in society with another best-selling book, The Third Wave, in 1980. In it, he foresaw such technological advances as cloning, personal computers, the Internet, cable television and mobile communication. His later focus, via their other best-seller, Powershift, (1990), was on the increasing power of 21st-century military hardware and the proliferation of new technologies.
Alvin Toffler	930		He founded Toffler Associates, a management consulting company, and was a visiting scholar at the Russell Sage Foundation, visiting professor at Cornell University, faculty member of the New School for Social Research, a White House correspondent, and a business consultant. Toffler's ideas and writings were a significant influence on the thinking of business and government leaders worldwide, including China's Zhao Ziyang, and AOL founder Steve Case.
Alvin Toffler	930	Early life	Alvin Toffler was born on October 4, 1928, in New York City, and raised in Brooklyn. He was the son of Rose (Albaum) and Sam Toffler, a furrier, both Polish Jews who had migrated to America. He had one younger sister. He was inspired to become a writer at the age of 7 by his aunt and uncle, who lived with the Tofflers. "They were Depression-era literary intellectuals," Toffler said, "and they always talked about exciting ideas."
Alvin Toffler	930	Early life	Toffler graduated from New York University in 1950 as an English major, though by his own account he was more focused on political activism than grades. He met his future wife, Adelaide Elizabeth Farrell (nicknamed "Heidi"), when she was starting a graduate course in linguistics. Being radical students, they decided against further graduate work and moved to Cleveland, Ohio, where they married on April 29, 1950.

A. A. Milne

924

Religious views

He wrote in the poem "Explained":

A. A. Milne

924

Religious views

Elizabeth Ann Said to her Nan: "Please will you tell me how God began? Somebody must have made Him. So Who could it be, 'cos I want to know?"

A. A. Milne

924

Religious views

He also wrote in the poem "Vespers":

A. A. Milne

924

Religious views

"Oh! Thank you, God, for a lovely day. And what was the other I had to say? I said "Bless Daddy," so what can it be? Oh! Now I remember it. God bless Me."

Asociación Alumni

925

Asociación Alumni, usually just Alumni, is an Argentine rugby union club located in Tortuguitas, Greater Buenos Aires. The senior squad currently competes at Top 12, the first division of the Unión de Rugby de Buenos Aires league system.

Asociación Alumni

925

The club has ties with former football club Alumni because both were established by Buenos Aires English High School students.

Asociación Alumni

925

History

The first club with the name "Alumni" played association football, having been found in 1898 by students of Buenos Aires English High School (BAEHS) along with director Alexander Watson Hutton. Originally under the name "English High School A.C.", the team would be later obliged by the Association to change its name, therefore "Alumni" was chosen, following a proposal by Carlos Bowers, a former student of the school.

Asociación Alumni

925

History

Alumni was the most successful team during the first years of Argentine football, winning 10 of 14 league championships contested. Alumni is still considered the first great football team in the country. Alumni was reorganised in 1908, "in order to encourage people to practise all kind of sports, specially football". This was the last try to develop itself as a sports club rather than just a football team, such as Lomas, Belgrano and Quilmes had successfully done in the past, but the efforts were not enough. Alumni played its last game in 1911 and was definitely dissolved on April 24, 1913.

Asociación Alumni

925

History

In 1951, two guards of the BAEHS, Daniel Ginhson (also a former player of Buenos Aires F.C.) and Guillermo Cubelli, supported by the school's alumni and fathers of the students, they decided to establish a club focused on rugby union exclusively. Former players still alive of Alumni football club and descendants of other players already dead gave their permission to use the name "Alumni".

Asociación Alumni

925

History

On December 13, in a meeting presided by Carlos Bowers himself (who had proposed the name "Alumni" to the original football team 50 years before), the club was officially established under the name "Asociación Juvenil Alumni", also adopting the same colors as its predecessor.

Asociación Alumni

925

History

The team achieved good results and in 1960 the club presented a team that won the third division of the Buenos Aires league, reaching the second division. Since then, Alumni has played at the highest level of Argentine rugby and its rivalry with Belgrano Athletic Club is one of the fiercest local derbies in Buenos Aires. Alumni would later climb up to first division winning 5 titles: 4 consecutive between 1989 and 1992, and the other in 2001.

Asociación Alumni

925

History

In 2002, Alumni won its first Nacional de Clubes title, defeating Jockey Club de Rosario 23–21 in the final.

Asociación Alumni

925

Players

As of January 2018:

Axiom

928

An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.

Axiom

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The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning.

Axiom

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In mathematics, an axiom may be a "logical axiom" or a "non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are substantive assertions about the elements of the domain of a specific mathematical theory, such as arithmetic.

Axiom

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Non-logical axioms may also be called "postulates" or "assumptions". In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., the parallel postulate in Euclidean geometry). To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms), and there are typically many ways to axiomatize a given mathematical domain.

Axiom

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Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the philosophy of mathematics.

Axiom

928

Etymology

The word axiom comes from the Greek word ἀξίωμα (axíōma), a verbal noun from the verb ἀξιόειν (axioein), meaning "to deem worthy", but also "to require", which in turn comes from ἄξιος (áxios), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the ancient Greek philosophers and mathematicians, axioms were taken to be immediately evident propositions, foundational and common to many fields of investigation, and self-evidently true without any further argument or proof.

Axiom

928

Etymology

The root meaning of the word postulate is to "demand"; for instance, Euclid demands that one agree that some things can be done (e.g., any two points can be joined by a straight line).

Axiom

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Etymology

Ancient geometers maintained some distinction between axioms and postulates. While commenting on Euclid's books, Proclus remarks that "Geminus held that this [4th] Postulate should not be classed as a postulate but as an axiom, since it does not, like the first three Postulates, assert the possibility of some construction but expresses an essential property." Boethius translated 'postulate' as petitio and called the axioms notiones communes but in later manuscripts this usage was not always strictly kept.

Axiom

928

Historical development

The logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments (syllogisms, rules of inference) was developed by the ancient Greeks, and has become the core principle of modern mathematics. Tautologies excluded, nothing can be deduced if nothing is assumed. Axioms and postulates are thus the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions (theorems, in the case of mathematics) must be proven with the aid of these basic assumptions. However, the interpretation of mathematical knowledge has changed from ancient times to the modern, and consequently the terms axiom and postulate hold a slightly different meaning for the present day mathematician, than they did for Aristotle and Euclid.

Axiom

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Historical development

The ancient Greeks considered geometry as just one of several sciences, and held the theorems of geometry on par with scientific facts. As such, they developed and used the logico-deductive method as a means of avoiding error, and for structuring and communicating knowledge. Aristotle's posterior analytics is a definitive exposition of the classical view.

Axiom

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Historical development

An "axiom", in classical terminology, referred to a self-evident assumption common to many branches of science. A good example would be the assertion that:

Axiom

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Historical development

When an equal amount is taken from equals, an equal amount results.

Axiom

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Historical development

At the foundation of the various sciences lay certain additional hypotheses that were accepted without proof. Such a hypothesis was termed a postulate. While the axioms were common to many sciences, the postulates of each particular science were different. Their validity had to be established by means of real-world experience. Aristotle warns that the content of a science cannot be successfully communicated if the learner is in doubt about the truth of the postulates.

Axiom

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Historical development

The classical approach is well-illustrated by Euclid's Elements, where a list of postulates is given (common-sensical geometric facts drawn from our experience), followed by a list of "common notions" (very basic, self-evident assertions).

Axiom

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Historical development

A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, propositions, theorems) and definitions. One must concede the need for primitive notions, or undefined terms or concepts, in any study. Such abstraction or formalization makes mathematical knowledge more general, capable of multiple different meanings, and therefore useful in multiple contexts. Alessandro Padoa, Mario Pieri, and Giuseppe Peano were pioneers in this movement.

Axiom

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Historical development

Structuralist mathematics goes further, and develops theories and axioms (e.g. field theory, group theory, topology, vector spaces) without any particular application in mind. The distinction between an "axiom" and a "postulate" disappears. The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts rests on the acceptance of the basic hypotheses. However, by throwing out Euclid's fifth postulate, one can get theories that have meaning in wider contexts (e.g., hyperbolic geometry). As such, one must simply be prepared to use labels such as "line" and "parallel" with greater flexibility. The development of hyperbolic geometry taught mathematicians that it is useful to regard postulates as purely formal statements, and not as facts based on experience.

Axiom

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Historical development

When mathematicians employ the field axioms, the intentions are even more abstract. The propositions of field theory do not concern any one particular application; the mathematician now works in complete abstraction. There are many examples of fields; field theory gives correct knowledge about them all.

Axiom

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Historical development

It is not correct to say that the axioms of field theory are "propositions that are regarded as true without proof." Rather, the field axioms are a set of constraints. If any given system of addition and multiplication satisfies these constraints, then one is in a position to instantly know a great deal of extra information about this system.

Axiom

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Historical development

Modern mathematics formalizes its foundations to such an extent that mathematical theories can be regarded as mathematical objects, and mathematics itself can be regarded as a branch of logic. Frege, Russell, Poincaré, Hilbert, and Gödel are some of the key figures in this development.

Axiom

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Historical development

Another lesson learned in modern mathematics is to examine purported proofs carefully for hidden assumptions.

Axiom

928

Historical development

In the modern understanding, a set of axioms is any collection of formally stated assertions from which other formally stated assertions follow – by the application of certain well-defined rules. In this view, logic becomes just another formal system. A set of axioms should be consistent; it should be impossible to derive a contradiction from the axioms. A set of axioms should also be non-redundant; an assertion that can be deduced from other axioms need not be regarded as an axiom.

Axiom

928

Historical development

It was the early hope of modern logicians that various branches of mathematics, perhaps all of mathematics, could be derived from a consistent collection of basic axioms. An early success of the formalist program was Hilbert's formalization of Euclidean geometry, and the related demonstration of the consistency of those axioms.

Axiom

928

Historical development

In a wider context, there was an attempt to base all of mathematics on Cantor's set theory. Here, the emergence of Russell's paradox and similar antinomies of naïve set theory raised the possibility that any such system could turn out to be inconsistent.

Axiom

928

Historical development

The formalist project suffered a decisive setback, when in 1931 Gödel showed that it is possible, for any sufficiently large set of axioms (Peano's axioms, for example) to construct a statement whose truth is independent of that set of axioms. As a corollary, Gödel proved that the consistency of a theory like Peano arithmetic is an unprovable assertion within the scope of that theory.

Axiom

928

Historical development

It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by the system of natural numbers, an infinite but intuitively accessible formal system. However, at present, there is no known way of demonstrating the consistency of the modern Zermelo–Fraenkel axioms for set theory. Furthermore, using techniques of forcing (Cohen) one can show that the continuum hypothesis (Cantor) is independent of the Zermelo–Fraenkel axioms. Thus, even this very general set of axioms cannot be regarded as the definitive foundation for mathematics.

Axiom

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Historical development

Experimental sciences - as opposed to mathematics and logic - also have general founding assertions from which a deductive reasoning can be built so as to express propositions that predict properties - either still general or much more specialized to a specific experimental context. For instance, Newton's laws in classical mechanics, Maxwell's equations in classical electromagnetism, Einstein's equation in general relativity, Mendel's laws of genetics, Darwin's Natural selection law, etc. These founding assertions are usually called principles or postulates so as to distinguish from mathematical axioms.

Axiom

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Historical development

As a matter of facts, the role of axioms in mathematics and postulates in experimental sciences is different. In mathematics one neither "proves" nor "disproves" an axiom. A set of mathematical axioms gives a set of rules that fix a conceptual realm, in which the theorems logically follow. In contrast, in experimental sciences, a set of postulates shall allow deducing results that match or do not match experimental results. If postulates do not allow deducing experimental predictions, they do not set a scientific conceptual framework and have to be completed or made more accurate. If the postulates allow deducing predictions of experimental results, the comparison with experiments allows falsifying (falsified) the theory that the postulates install. A theory is considered valid as long as it has not been falsified.

Axiom

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Historical development

Now, the transition between the mathematical axioms and scientific postulates is always slightly blurred, especially in physics. This is due to the heavy use of mathematical tools to support the physical theories. For instance, the introduction of Newton's laws rarely establishes as a prerequisite neither Euclidean geometry or differential calculus that they imply. It became more apparent when Albert Einstein first introduced special relativity where the invariant quantity is no more the Euclidean length l {\displaystyle l} (defined as l 2 = x 2 + y 2 + z 2 {\displaystyle l^{2}=x^{2}+y^{2}+z^{2}} ) > but the Minkowski spacetime interval s {\displaystyle s} (defined as s 2 = c 2 t 2 − x 2 − y 2 − z 2 {\displaystyle s^{2}=c^{2}t^{2}-x^{2}-y^{2}-z^{2}} ), and then general relativity where flat Minkowskian geometry is replaced with pseudo-Riemannian geometry on curved manifolds.

Axiom

928

Historical development

In quantum physics, two sets of postulates have coexisted for some time, which provide a very nice example of falsification. The 'Copenhagen school' (Niels Bohr, Werner Heisenberg, Max Born) developed an operational approach with a complete mathematical formalism that involves the description of quantum system by vectors ('states') in a separable Hilbert space, and physical quantities as linear operators that act in this Hilbert space. This approach is fully falsifiable and has so far produced the most accurate predictions in physics. But it has the unsatisfactory aspect of not allowing answers to questions one would naturally ask. For this reason, another 'hidden variables' approach was developed for some time by Albert Einstein, Erwin Schrödinger, David Bohm. It was created so as to try to give deterministic explanation to phenomena such as entanglement. This approach assumed that the Copenhagen school description was not complete, and postulated that some yet unknown variable was to be added to the theory so as to allow answering some of the questions it does not answer (the founding elements of which were discussed as the EPR paradox in 1935). Taking this ideas seriously, John Bell derived in 1964 a prediction that would lead to different experimental results (Bell's inequalities) in the Copenhagen and the Hidden variable case. The experiment was conducted first by Alain Aspect in the early 1980's, and the result excluded the simple hidden variable approach (sophisticated hidden variables could still exist but their properties would still be more disturbing than the problems they try to solve). This does not mean that the conceptual framework of quantum physics can be considered as complete now, since some open questions still exist (the limit between the quantum and classical realms, what happens during a quantum measurement, what happens in a completely closed quantum system such as the universe itself, etc.).

Axiom

928

Mathematical logic

In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical and non-logical (somewhat similar to the ancient distinction between "axioms" and "postulates" respectively).

Axiom

928

Mathematical logic

These are certain formulas in a formal language that are universally valid, that is, formulas that are satisfied by every assignment of values. Usually one takes as logical axioms at least some minimal set of tautologies that is sufficient for proving all tautologies in the language; in the case of predicate logic more logical axioms than that are required, in order to prove logical truths that are not tautologies in the strict sense.

Axiom

928

Mathematical logic

In propositional logic it is common to take as logical axioms all formulae of the following forms, where ϕ {\displaystyle \phi } , χ {\displaystyle \chi } , and ψ {\displaystyle \psi } can be any formulae of the language and where the included primitive connectives are only " ¬ {\displaystyle \neg } " for negation of the immediately following proposition and " → {\displaystyle \to } " for a preconceptions=sequence of forthcoming events of times of moment taking place. implication from antecedent to consequent propositions:

Axiom

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Mathematical logic

Each of these patterns is an axiom schema, a rule for generating an infinite number of axioms. For example, if A {\displaystyle A} , B {\displaystyle B} , and C {\displaystyle C} are propositional variables, then A → ( B → A ) {\displaystyle A\to (B\to A)} and ( A → ¬ B ) → ( C → ( A → ¬ B ) ) {\displaystyle (A\to \lnot B)\to (C\to (A\to \lnot B))} are both instances of axiom schema 1, and hence are axioms. It can be shown that with only these three axiom schemata and modus ponens, one can prove all tautologies of the propositional calculus. It can also be shown that no pair of these schemata is sufficient for proving all tautologies with modus ponens.

Axiom

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Mathematical logic

Other axiom schemata involving the same or different sets of primitive connectives can be alternatively constructed.

Axiom

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Mathematical logic

These axiom schemata are also used in the predicate calculus, but additional logical axioms are needed to include a quantifier in the calculus.

Axiom

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Mathematical logic

Axiom of Equality.Let L {\displaystyle {\mathfrak {L}}} be a first-order language. For each variable x {\displaystyle x} , the below formula is universally valid.

Axiom

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Mathematical logic

x = x {\displaystyle x=x}

Axiom

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Mathematical logic

This means that, for any variable symbol x {\displaystyle x} , the formula x = x {\displaystyle x=x} can be regarded as an axiom. Also, in this example, for this not to fall into vagueness and a never-ending series of "primitive notions", either a precise notion of what we mean by x = x {\displaystyle x=x} (or, for that matter, "to be equal") has to be well established first, or a purely formal and syntactical usage of the symbol = {\displaystyle =} has to be enforced, only regarding it as a string and only a string of symbols, and mathematical logic does indeed do that.

Axiom

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Mathematical logic

Another, more interesting example axiom scheme, is that which provides us with what is known as Universal Instantiation:

Axiom

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Mathematical logic

Axiom scheme for Universal Instantiation.Given a formula ϕ {\displaystyle \phi } in a first-order language L {\displaystyle {\mathfrak {L}}} , a variable x {\displaystyle x} and a term t {\displaystyle t} that is substitutable for x {\displaystyle x} in ϕ {\displaystyle \phi } , the below formula is universally valid.

Axiom

928

Mathematical logic

∀ x ϕ → ϕ t x {\displaystyle \forall x\,\phi \to \phi _{t}^{x}}

Axiom

928

Mathematical logic

Where the symbol ϕ t x {\displaystyle \phi _{t}^{x}} stands for the formula ϕ {\displaystyle \phi } with the term t {\displaystyle t} substituted for x {\displaystyle x} . (See Substitution of variables.) In informal terms, this example allows us to state that, if we know that a certain property P {\displaystyle P} holds for every x {\displaystyle x} and that t {\displaystyle t} stands for a particular object in our structure, then we should be able to claim P ( t ) {\displaystyle P(t)} . Again, we are claiming that the formula ∀ x ϕ → ϕ t x {\displaystyle \forall x\phi \to \phi _{t}^{x}} is valid, that is, we must be able to give a "proof" of this fact, or more properly speaking, a metaproof. These examples are metatheorems of our theory of mathematical logic since we are dealing with the very concept of proof itself. Aside from this, we can also have Existential Generalization:

Axiom

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Mathematical logic

Axiom scheme for Existential Generalization. Given a formula ϕ {\displaystyle \phi } in a first-order language L {\displaystyle {\mathfrak {L}}} , a variable x {\displaystyle x} and a term t {\displaystyle t} that is substitutable for x {\displaystyle x} in ϕ {\displaystyle \phi } , the below formula is universally valid.

Axiom

928

Mathematical logic

ϕ t x → ∃ x ϕ {\displaystyle \phi _{t}^{x}\to \exists x\,\phi }

Axiom

928

Mathematical logic

Non-logical axioms are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example, the natural numbers and the integers, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as groups). Thus non-logical axioms, unlike logical axioms, are not tautologies. Another name for a non-logical axiom is postulate.

Axiom

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Mathematical logic

Almost every modern mathematical theory starts from a given set of non-logical axioms, and it was thought that, in principle, every theory could be axiomatized in this way and formalized down to the bare language of logical formulas.

Axiom

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Mathematical logic

Non-logical axioms are often simply referred to as axioms in mathematical discourse. This does not mean that it is claimed that they are true in some absolute sense. For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom, we can do quite well developing (the more general) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.

Axiom

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Mathematical logic

Thus, an axiom is an elementary basis for a formal logic system that together with the rules of inference define a deductive system.

Axiom

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Mathematical logic

This section gives examples of mathematical theories that are developed entirely from a set of non-logical axioms (axioms, henceforth). A rigorous treatment of any of these topics begins with a specification of these axioms.

Axiom

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Mathematical logic

Basic theories, such as arithmetic, real analysis and complex analysis are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of Zermelo–Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann–Bernays–Gödel set theory, a conservative extension of ZFC. Sometimes slightly stronger theories such as Morse–Kelley set theory or set theory with a strongly inaccessible cardinal allowing the use of a Grothendieck universe is used, but in fact, most mathematicians can actually prove all they need in systems weaker than ZFC, such as second-order arithmetic.

Axiom

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Mathematical logic

The study of topology in mathematics extends all over through point set topology, algebraic topology, differential topology, and all the related paraphernalia, such as homology theory, homotopy theory. The development of abstract algebra brought with itself group theory, rings, fields, and Galois theory.

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Mathematical logic

This list could be expanded to include most fields of mathematics, including measure theory, ergodic theory, probability, representation theory, and differential geometry.

Axiom

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Mathematical logic

The Peano axioms are the most widely used axiomatization of first-order arithmetic. They are a set of axioms strong enough to prove many important facts about number theory and they allowed Gödel to establish his famous second incompleteness theorem.

Axiom

928

Mathematical logic

We have a language L N T = { 0 , S } {\displaystyle {\mathfrak {L}}_{NT}=\{0,S\}} where 0 {\displaystyle 0} is a constant symbol and S {\displaystyle S} is a unary function and the following axioms:

Axiom

928

Mathematical logic

The standard structure is N = ⟨ N , 0 , S ⟩ {\displaystyle {\mathfrak {N}}=\langle \mathbb {N} ,0,S\rangle } where N {\displaystyle \mathbb {N} } is the set of natural numbers, S {\displaystyle S} is the successor function and 0 {\displaystyle 0} is naturally interpreted as the number 0.

Axiom

928

Mathematical logic

Probably the oldest, and most famous, list of axioms are the 4 + 1 Euclid's postulates of plane geometry. The axioms are referred to as "4 + 1" because for nearly two millennia the fifth (parallel) postulate ("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four. Ultimately, the fifth postulate was found to be independent of the first four. One can assume that exactly one parallel through a point outside a line exists, or that infinitely many exist. This choice gives us two alternative forms of geometry in which the interior angles of a triangle add up to exactly 180 degrees or less, respectively, and are known as Euclidean and hyperbolic geometries. If one also removes the second postulate ("a line can be extended indefinitely") then elliptic geometry arises, where there is no parallel through a point outside a line, and in which the interior angles of a triangle add up to more than 180 degrees.

Axiom

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Mathematical logic

The objectives of the study are within the domain of real numbers. The real numbers are uniquely picked out (up to isomorphism) by the properties of a Dedekind complete ordered field, meaning that any nonempty set of real numbers with an upper bound has a least upper bound. However, expressing these properties as axioms requires the use of second-order logic. The Löwenheim–Skolem theorems tell us that if we restrict ourselves to first-order logic, any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger. Some of the latter are studied in non-standard analysis.

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Mathematical logic

A deductive system consists of a set Λ {\displaystyle \Lambda } of logical axioms, a set Σ {\displaystyle \Sigma } of non-logical axioms, and a set { ( Γ , ϕ ) } {\displaystyle \{(\Gamma ,\phi )\}} of rules of inference. A desirable property of a deductive system is that it be complete. A system is said to be complete if, for all formulas ϕ {\displaystyle \phi } ,

Axiom

928

Mathematical logic

if Σ ⊨ ϕ then Σ ⊢ ϕ {\displaystyle {\text{if }}\Sigma \models \phi {\text{ then }}\Sigma \vdash \phi }

Axiom

928

Mathematical logic

that is, for any statement that is a logical consequence of Σ {\displaystyle \Sigma } there actually exists a deduction of the statement from Σ {\displaystyle \Sigma } . This is sometimes expressed as "everything that is true is provable", but it must be understood that "true" here means "made true by the set of axioms", and not, for example, "true in the intended interpretation". Gödel's completeness theorem establishes the completeness of a certain commonly used type of deductive system.

Axiom

928

Mathematical logic

Note that "completeness" has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms Σ {\displaystyle \Sigma } of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement ϕ {\displaystyle \phi } such that neither ϕ {\displaystyle \phi } nor ¬ ϕ {\displaystyle \lnot \phi } can be proved from the given set of axioms.

Axiom

928

Mathematical logic

There is thus, on the one hand, the notion of completeness of a deductive system and on the other hand that of completeness of a set of non-logical axioms. The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another.

Axiom

928

Mathematical logic

Early mathematicians regarded axiomatic geometry as a model of physical space, and obviously, there could only be one such model. The idea that alternative mathematical systems might exist was very troubling to mathematicians of the 19th century and the developers of systems such as Boolean algebra made elaborate efforts to derive them from traditional arithmetic. Galois showed just before his untimely death that these efforts were largely wasted. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details, and modern algebra was born. In the modern view, axioms may be any set of formulas, as long as they are not known to be inconsistent.

Alpha

929

Alpha /ˈælfə/ (uppercase Α, lowercase α; Ancient Greek: ἄλφα, álpha, or Greek: άλφα, romanized: álfa) is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter aleph , which is the West Semitic word for "ox". Letters that arose from alpha include the Latin letter A and the Cyrillic letter А.

Alpha

929

Uses

In Ancient Greek, alpha was pronounced [a] and could be either phonemically long ([aː]) or short ([a]). Where there is ambiguity, long and short alpha are sometimes written with a macron and breve today: Ᾱᾱ, Ᾰᾰ.

Alpha

929

Uses

In Modern Greek, vowel length has been lost, and all instances of alpha simply represent the open front unrounded vowel IPA: [a].

Alpha

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Uses

In the polytonic orthography of Greek, alpha, like other vowel letters, can occur with several diacritic marks: any of three accent symbols (ά, ὰ, ᾶ), and either of two breathing marks (ἁ, ἀ), as well as combinations of these. It can also combine with the iota subscript (ᾳ).

Alpha

929

Uses

In the Attic–Ionic dialect of Ancient Greek, long alpha [aː] fronted to [ɛː] (eta). In Ionic, the shift took place in all positions. In Attic, the shift did not take place after epsilon, iota, and rho (ε, ι, ρ; e, i, r). In Doric and Aeolic, long alpha is preserved in all positions.

Alpha

929

Uses

Privative a is the Ancient Greek prefix ἀ- or ἀν- a-, an-, added to words to negate them. It originates from the Proto-Indo-European *n̥- (syllabic nasal) and is cognate with English un-.

Alpha

929

Uses

Copulative a is the Greek prefix ἁ- or ἀ- ha-, a-. It comes from Proto-Indo-European *sm̥.

Alpha

929

Uses

The letter alpha represents various concepts in physics and chemistry, including alpha radiation, angular acceleration, alpha particles, alpha carbon and strength of electromagnetic interaction (as fine-structure constant). Alpha also stands for thermal expansion coefficient of a compound in physical chemistry. It is also commonly used in mathematics in algebraic solutions representing quantities such as angles. Furthermore, in mathematics, the letter alpha is used to denote the area underneath a normal curve in statistics to denote significance level when proving null and alternative hypotheses. In ethology, it is used to name the dominant individual in a group of animals. In aerodynamics, the letter is used as a symbol for the angle of attack of an aircraft and the word "alpha" is used as a synonym for this property. In mathematical logic, α is sometimes used as a placeholder for ordinal numbers.

Alpha

929

Uses

The proportionality operator "∝" (in Unicode: U+221D) is sometimes mistaken for alpha.

Alpha

929

Uses

The uppercase letter alpha is not generally used as a symbol because it tends to be rendered identically to the uppercase Latin A.

Alpha

929

Uses

In the International Phonetic Alphabet, the letter ɑ, which looks similar to the lower-case alpha, represents the open back unrounded vowel.

Alpha

929

History and symbolism

The Phoenician alphabet was adopted for Greek in the early 8th century BC, perhaps in Euboea. The majority of the letters of the Phoenician alphabet were adopted into Greek with much the same sounds as they had had in Phoenician, but ʼāleph, the Phoenician letter representing the glottal stop [ʔ], was adopted as representing the vowel [a]; similarly, hē [h] and ʽayin [ʕ] are Phoenician consonants that became Greek vowels, epsilon [e] and omicron [o], respectively.

Alpha

929

History and symbolism

Plutarch, in Moralia, presents a discussion on why the letter alpha stands first in the alphabet. Ammonius asks Plutarch what he, being a Boeotian, has to say for Cadmus, the Phoenician who reputedly settled in Thebes and introduced the alphabet to Greece, placing alpha first because it is the Phoenician name for ox—which, unlike Hesiod, the Phoenicians considered not the second or third, but the first of all necessities. "Nothing at all," Plutarch replied. He then added that he would rather be assisted by Lamprias, his own grandfather, than by Dionysus' grandfather, i.e. Cadmus. For Lamprias had said that the first articulate sound made is "alpha", because it is very plain and simple—the air coming off the mouth does not require any motion of the tongue—and therefore this is the first sound that children make.

Alpha

929

History and symbolism

According to Plutarch's natural order of attribution of the vowels to the planets, alpha was connected with the Moon.

Alpha

929

History and symbolism

As the first letter of the alphabet, Alpha as a Greek numeral came to represent the number 1. Therefore, Alpha, both as a symbol and term, is used to refer to the "first", or "primary", or "principal" (most significant) occurrence or status of a thing.

Alpha

929

History and symbolism

The New Testament has God declaring himself to be the "Alpha and Omega, the beginning and the end, the first and the last." (Revelation 22:13, KJV, and see also 1:8).

Alpha

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History and symbolism

Consequently, the term "alpha" has also come to be used to denote "primary" position in social hierarchy, examples being the concept of dominant "alpha" members in groups of animals.

Alpha

929

Computer encodings

For accented Greek characters, see Greek diacritics: Computer encoding.

Alvin Toffler

930

Alvin Eugene Toffler (October 4, 1928 – June 27, 2016) was an American writer, futurist, and businessman known for his works discussing modern technologies, including the digital revolution and the communication revolution, with emphasis on their effects on cultures worldwide. He is regarded as one of the world's outstanding futurists.

Alvin Toffler

930

Toffler was an associate editor of Fortune magazine. In his early works he focused on technology and its impact, which he termed "information overload". In 1970, his first major book about the future, Future Shock, became a worldwide best-seller and has sold over 6 million copies.

Alvin Toffler

930

He and his wife Heidi Toffler, who collaborated with him for most of his writings, moved on to examining the reaction to changes in society with another best-selling book, The Third Wave, in 1980. In it, he foresaw such technological advances as cloning, personal computers, the Internet, cable television and mobile communication. His later focus, via their other best-seller, Powershift, (1990), was on the increasing power of 21st-century military hardware and the proliferation of new technologies.

Alvin Toffler

930

He founded Toffler Associates, a management consulting company, and was a visiting scholar at the Russell Sage Foundation, visiting professor at Cornell University, faculty member of the New School for Social Research, a White House correspondent, and a business consultant. Toffler's ideas and writings were a significant influence on the thinking of business and government leaders worldwide, including China's Zhao Ziyang, and AOL founder Steve Case.

Alvin Toffler

930

Early life

Alvin Toffler was born on October 4, 1928, in New York City, and raised in Brooklyn. He was the son of Rose (Albaum) and Sam Toffler, a furrier, both Polish Jews who had migrated to America. He had one younger sister. He was inspired to become a writer at the age of 7 by his aunt and uncle, who lived with the Tofflers. "They were Depression-era literary intellectuals," Toffler said, "and they always talked about exciting ideas."

Alvin Toffler

930

Early life

Toffler graduated from New York University in 1950 as an English major, though by his own account he was more focused on political activism than grades. He met his future wife, Adelaide Elizabeth Farrell (nicknamed "Heidi"), when she was starting a graduate course in linguistics. Being radical students, they decided against further graduate work and moved to Cleveland, Ohio, where they married on April 29, 1950.