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7,400 | In mathematical logic, an (induced) substructure or (induced) subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are restricted to the substructure's domain. Some examples of subalgebras are subgroups, submonoids, subrings, subfields, subalgebras of algebras over a field, or induced subgraphs. Shifting the point of view, the larger structure is called an extension or a superstructure of its substructure | Substructure (mathematics) |
7,401 | In mathematical logic, Tarski's high school algebra problem was a question posed by Alfred Tarski. It asks whether there are identities involving addition, multiplication, and exponentiation over the positive integers that cannot be proved using eleven axioms about these operations that are taught in high-school-level mathematics. The question was solved in 1980 by Alex Wilkie, who showed that such unprovable identities do exist | Tarski's high school algebra problem |
7,402 | In universal algebra and mathematical logic, a term algebra is a freely generated algebraic structure over a given signature. For example, in a signature consisting of a single binary operation, the term algebra over a set X of variables is exactly the free magma generated by X. Other synonyms for the notion include absolutely free algebra and anarchic algebra | Term algebra |
7,403 | In universal algebra and lattice theory, a tolerance relation on an algebraic structure is a reflexive symmetric relation that is compatible with all operations of the structure. Thus a tolerance is like a congruence, except that the assumption of transitivity is dropped. On a set, an algebraic structure with empty family of operations, tolerance relations are simply reflexive symmetric relations | Tolerance relation |
7,404 | The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature | Ultraproduct |
7,405 | In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras, and (direct) products | Variety (universal algebra) |
7,406 | The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points | Banach–Tarski paradox |
7,407 | In mathematical logic, the Barwise compactness theorem, named after Jon Barwise, is a generalization of the usual compactness theorem for first-order logic to a certain class of infinitary languages. It was stated and proved by Barwise in 1967.
Statement
Let
A
{\displaystyle A}
be a countable admissible set | Barwise compactness theorem |
7,408 | In mathematical logic, Beth definability is a result that connects implicit definability of a property to its explicit definability. Specifically Beth definability states that the two senses of definability are equivalent.
First-order logic has the Beth definability property | Beth definability |
7,409 | In descriptive set theory, the Borel determinacy theorem states that any Gale–Stewart game whose payoff set is a Borel set is determined, meaning that one of the two players will have a winning strategy for the game. A Gale-Stewart game is a possibly infinite two-player game, where both players have perfect information and no randomness is involved.
The theorem is a far reaching generalization of Zermelo's Theorem about the determinacy of finite games | Borel determinacy theorem |
7,410 | In mathematics, the Bourbaki–Witt theorem in order theory, named after Nicolas Bourbaki and Ernst Witt, is a basic fixed point theorem for partially ordered sets. It states that if X is a non-empty chain complete poset, and
f
:
X
→
X
{\displaystyle f:X\to X}
such that
f
(
x
)
≥
x
{\displaystyle f(x)\geq x}
for all
x
,
{\displaystyle x,}
then f has a fixed point. Such a function f is called inflationary or progressive | Bourbaki–Witt theorem |
7,411 | In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. : 20– Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began.
The diagonal argument was not Cantor's first proof of the uncountability of the real numbers, which appeared in 1874 | Cantor's diagonal argument |
7,412 | In order theory and model theory, branches of mathematics, Cantor's isomorphism theorem states that every two countable dense unbounded linear orders are order-isomorphic. For instance, Minkowski's question-mark function produces an isomorphism (a one-to-one order-preserving correspondence) between the numerical ordering of the rational numbers and the numerical ordering of the dyadic rationals.
The theorem is named after Georg Cantor, who first published it in 1895, using it to characterize the (uncountable) ordering on the real numbers | Cantor's isomorphism theorem |
7,413 | In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set
A
{\displaystyle A}
, the set of all subsets of
A
,
{\displaystyle A,}
the power set of
A
,
{\displaystyle A,}
has a strictly greater cardinality than
A
{\displaystyle A}
itself.
For finite sets, Cantor's theorem can be seen to be true by simple enumeration of the number of subsets. Counting the empty set as a subset, a set with
n
{\displaystyle n}
elements has a total of
2
n
{\displaystyle 2^{n}}
subsets, and the theorem holds because
2
n
>
n
{\displaystyle 2^{n}>n}
for all non-negative integers | Cantor's theorem |
7,414 | In mathematical logic, a theory is categorical if it has exactly one model (up to isomorphism). Such a theory can be viewed as defining its model, uniquely characterizing the model's structure.
In first-order logic, only theories with a finite model can be categorical | Categorical theory |
7,415 | In lambda calculus, the Church–Rosser theorem states that, when applying reduction rules to terms, the ordering in which the reductions are chosen does not make a difference to the eventual result.
More precisely, if there are two distinct reductions or sequences of reductions that can be applied to the same term, then there exists a term that is reachable from both results, by applying (possibly empty) sequences of additional reductions. The theorem was proved in 1936 by Alonzo Church and J | Church–Rosser theorem |
7,416 | Codd's theorem states that relational algebra and the domain-independent relational calculus queries, two well-known foundational query languages for the relational model, are precisely equivalent in expressive power. That is, a database query can be formulated in one language if and only if it can be expressed in the other.
The theorem is named after Edgar F | Codd's theorem |
7,417 | In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally not effective) method for constructing models of any set of sentences that is finitely consistent.
The compactness theorem for the propositional calculus is a consequence of Tychonoff's theorem (which says that the product of compact spaces is compact) applied to compact Stone spaces, hence the theorem's name | Compactness theorem |
7,418 | In sequent calculus, the completeness of atomic initial sequents states that initial sequents A ⊢ A (where A is an arbitrary formula) can be derived from only atomic initial sequents p ⊢ p (where p is an atomic formula). This theorem plays a role analogous to eta expansion in lambda calculus, and dual to cut-elimination and beta reduction. Typically it can be established by induction on the structure of A, much more easily than cut-elimination | Completeness of atomic initial sequents |
7,419 | In computational complexity theory, the compression theorem is an important theorem about the complexity of computable functions.
The theorem states that there exists no largest complexity class, with computable boundary, which contains all computable functions.
Compression theorem
Given a Gödel numbering
φ
{\displaystyle \varphi }
of the computable functions and a Blum complexity measure
Φ
{\displaystyle \Phi }
where a complexity class for a boundary function
f
{\displaystyle f}
is defined as
C
(
f
)
:=
{
φ
i
∈
R
(
1
)
|
(
∀
∞
x
)
Φ
i
(
x
)
≤
f
(
x
)
}
| Compression theorem |
7,420 | In mathematical logic, Craig's theorem states that any recursively enumerable set of well-formed formulas of a first-order language is (primitively) recursively axiomatizable. This result is not related to the well-known Craig interpolation theorem, although both results are named after the same logician, William Craig.
Recursive axiomatization
Let
A
1
,
A
2
,
…
{\displaystyle A_{1},A_{2},\dots }
be an enumeration of the axioms of a recursively enumerable set T of first-order formulas | Craig's theorem |
7,421 | The cut-elimination theorem (or Gentzen's Hauptsatz) is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen in his landmark 1934 paper "Investigations in Logical Deduction" for the systems LJ and LK formalising intuitionistic and classical logic respectively. The cut-elimination theorem states that any judgement that possesses a proof in the sequent calculus making use of the cut rule also possesses a cut-free proof, that is, a proof that does not make use of the cut rule | Cut-elimination theorem |
7,422 | In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly axiomatize that hypothesis, i. e. to prove an implication A → B, it is sufficient to assume A as an hypothesis and then proceed to derive B | Deduction theorem |
7,423 | In set theory, Easton's theorem is a result on the possible cardinal numbers of powersets. Easton (1970) (extending a result of Robert M. Solovay) showed via forcing that the only constraints on permissible values for 2κ when κ is a regular cardinal are
κ
<
cf
(
2
κ
)
{\displaystyle \kappa <\operatorname {cf} (2^{\kappa })}
(where cf(α) is the cofinality of α) and
if
κ
<
λ
then
2
κ
≤
2
λ
| Easton's theorem |
7,424 | In mathematical logic, a theory can be extended with
new constants or function names under certain conditions with assurance that the extension will introduce
no contradiction. Extension by definitions is perhaps the best-known approach, but it requires
unique existence of an object with the desired property. Addition of new names can also be done
safely without uniqueness | Extension by new constant and function names |
7,425 | In metalogic and metamathematics, Frege's theorem is a metatheorem that states that the Peano axioms of arithmetic can be derived in second-order logic from Hume's principle. It was first proven, informally, by Gottlob Frege in his 1884 Die Grundlagen der Arithmetik (The Foundations of Arithmetic) and proven more formally in his 1893 Grundgesetze der Arithmetik I (Basic Laws of Arithmetic I). The theorem was re-discovered by Crispin Wright in the early 1980s and has since been the focus of significant work | Frege's theorem |
7,426 | In mathematics, Gödel's speed-up theorem, proved by Gödel (1936), shows that there are theorems whose proofs can be drastically shortened by working in more powerful axiomatic systems.
Kurt Gödel showed how to find explicit examples of statements in formal systems that are provable in that system but whose shortest proof is unimaginably long. For example, the statement:
"This statement cannot be proved in Peano arithmetic in fewer than a googolplex symbols"is provable in Peano arithmetic (PA) but the shortest proof has at least a googolplex symbols, by an argument similar to the proof of Gödel's first incompleteness theorem: If PA is consistent, then it cannot prove the statement in fewer than a googolplex symbols, because the existence of such a proof would itself be a theorem of PA, a contradiction | Gödel's speed-up theorem |
7,427 | Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic.
The completeness theorem applies to any first-order theory: If T is such a theory, and φ is a sentence (in the same language) and every model of T is a model of φ, then there is a (first-order) proof of φ using the statements of T as axioms. One sometimes says this as "anything universally true is provable" | Gödel's completeness theorem |
7,428 | Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible | Gödel's incompleteness theorems |
7,429 | In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence eventually terminates at 0. Laurence Kirby and Jeff Paris showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second-order arithmetic). This was the third example of a true statement that is unprovable in Peano arithmetic, after the examples provided by Gödel's incompleteness theorem and Gerhard Gentzen's 1943 direct proof of the unprovability of ε0-induction in Peano arithmetic | Goodstein's theorem |
7,430 | In mathematics, the Halpern–Läuchli theorem is a partition result about finite products of infinite trees. Its original purpose was to give a model for set theory in which the Boolean prime ideal theorem is true but the axiom of choice is false. It is often called the Halpern–Läuchli theorem, but the proper attribution for the theorem as it is formulated below is to Halpern–Läuchli–Laver–Pincus or HLLP (named after James D | Halpern–Läuchli theorem |
7,431 | Herbrand's theorem is a fundamental result of mathematical logic obtained by Jacques Herbrand (1930). It essentially allows a certain kind of reduction of first-order logic to propositional logic. Although Herbrand originally proved his theorem for arbitrary formulas of first-order logic, the simpler version shown here, restricted to formulas in prenex form containing only existential quantifiers, became more popular | Herbrand's theorem |
7,432 | In mathematical logic, the Kanamori–McAloon theorem, due to Kanamori & McAloon (1987), gives an example of an incompleteness in Peano arithmetic, similar to that of the Paris–Harrington theorem. They showed that a certain finitistic theorem in Ramsey theory is not provable in Peano arithmetic (PA).
Statement
Given a set
s
⊆
N
{\displaystyle s\subseteq \mathbb {N} }
of non-negative integers, let
min
(
s
)
{\displaystyle \min(s)}
denote the minimum element of
s
{\displaystyle s}
| Kanamori–McAloon theorem |
7,433 | In computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions. The theorems were first proved by Stephen Kleene in 1938 and appear in his 1952 book Introduction to Metamathematics. A related theorem, which constructs fixed points of a computable function, is known as Rogers's theorem and is due to Hartley Rogers, Jr | Kleene's recursion theorem |
7,434 | In the mathematical areas of order and lattice theory, the Knaster–Tarski theorem, named after Bronisław Knaster and Alfred Tarski, states the following:
Let (L, ≤) be a complete lattice and let f : L → L be an monotonic function (w. r. t | Knaster–Tarski theorem |
7,435 | In set theory, König's theorem states that if the axiom of choice holds, I is a set,
κ
i
{\displaystyle \kappa _{i}}
and
λ
i
{\displaystyle \lambda _{i}}
are cardinal numbers for every i in I, and
κ
i
<
λ
i
{\displaystyle \kappa _{i}<\lambda _{i}}
for every i in I, then
∑
i
∈
I
κ
i
<
∏
i
∈
I
λ
i
.
{\displaystyle \sum _{i\in I}\kappa _{i}<\prod _{i\in I}\lambda _{i}. }
The sum here is the cardinality of the disjoint union of the sets mi, and the product is the cardinality of the Cartesian product | König's theorem (set theory) |
7,436 | In mathematical logic, Lindström's theorem (named after Swedish logician Per Lindström, who published it in 1969) states that first-order logic is the strongest logic (satisfying certain conditions, e. g. closure under classical negation) having both the (countable) compactness property and the (downward) Löwenheim–Skolem property | Lindström's theorem |
7,437 | In mathematical logic, Löb's theorem states that in Peano arithmetic (PA) (or any formal system including PA), for any formula P, if it is provable in PA that "if P is provable in PA then P is true", then P is provable in PA. If Prov(P) means that the formula P is provable, we may express this more formally as
If
P
A
⊢
P
r
o
v
(
P
)
→
P
{\displaystyle {\mathit {PA}}\vdash {\mathrm {Prov} (P)\rightarrow P}}
then
P
A
⊢
P
{\displaystyle {\mathit {PA}}\vdash P}
. An immediate corollary (the contrapositive) of Löb's theorem is that, if P is not provable in PA, then "if P is provable in PA, then P is true" is not provable in PA | Löb's theorem |
7,438 | In model theory, a branch of mathematical logic, the Łoś–Vaught test is a criterion for a theory to be complete, unable to be augmented without becoming inconsistent. For theories in classical logic, this means that for every sentence, the theory contains either the sentence or its negation but not both.
Statement
A theory
T
{\displaystyle T}
with signature σ is
κ
{\displaystyle \kappa }
-categorical for an infinite cardinal
κ
{\displaystyle \kappa }
if
T
{\displaystyle T}
has exactly one model (up to isomorphism) of cardinality
κ
| Łoś–Vaught test |
7,439 | In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem.
The precise formulation is given below. It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ, and that no first-order theory with an infinite model can have a unique model up to isomorphism | Löwenheim–Skolem theorem |
7,440 | In descriptive set theory and mathematical logic, Lusin's separation theorem states that if A and B are disjoint analytic subsets of Polish space, then there is a Borel set C in the space such that A ⊆ C and B ∩ C = ∅. It is named after Nikolai Luzin, who proved it in 1927. The theorem can be generalized to show that for each sequence (An) of disjoint analytic sets there is a sequence (Bn) of disjoint Borel sets such that An ⊆ Bn for each n | Lusin's separation theorem |
7,441 | In mathematical logic, the Paris–Harrington theorem states that a certain combinatorial principle in Ramsey theory, namely the strengthened finite Ramsey theorem, which is expressible in Peano arithmetic, is not provable in this system. The combinatorial principle is however provable in slightly stronger systems.
This result has been described by some (such as the editor of the Handbook of Mathematical Logic in the references below) as the first "natural" example of a true statement about the integers that could be stated in the language of arithmetic, but not proved in Peano arithmetic; it was already known that such statements existed by Gödel's first incompleteness theorem | Paris–Harrington theorem |
7,442 | In computability theory Post's theorem, named after Emil Post, describes the connection between the arithmetical hierarchy and the Turing degrees.
Background
The statement of Post's theorem uses several concepts relating to definability and recursion theory. This section gives a brief overview of these concepts, which are covered in depth in their respective articles | Post's theorem |
7,443 | In computability theory, the Rice–Shapiro theorem is a generalization of Rice's theorem, and is named after Henry Gordon Rice and Norman Shapiro.
Formal statement
Let A be a set of partial-recursive unary functions on the domain of natural numbers such that the set
I
x
(
A
)
:=
{
n
∣
φ
n
∈
A
}
{\displaystyle Ix(A):=\{n\mid \varphi _{n}\in A\}}
is recursively enumerable, where
φ
n
{\displaystyle \varphi _{n}}
denotes the
n
{\displaystyle n}
-th partial-recursive function in a Gödel numbering.
Then for any unary partial-recursive function
ψ
{\displaystyle \psi }
, we have:
ψ
∈
A
⇔
∃
{\displaystyle \psi \in A\Leftrightarrow \exists }
a finite function
θ
⊆
ψ
{\displaystyle \theta \subseteq \psi }
such that
θ
∈
A
| Rice–Shapiro theorem |
7,444 | In computability theory, Rice's theorem states that all non-trivial semantic properties of programs are undecidable. A semantic property is one about the program's behavior (for instance, does the program terminate for all inputs), unlike a syntactic property (for instance, does the program contain an if-then-else statement). A property is non-trivial if it is neither true for every partial computable function, nor false for every partial computable function | Rice's theorem |
7,445 | In mathematics, Richardson's theorem establishes the undecidability of the equality of real numbers defined by expressions involving integers, π,
ln
2
,
{\displaystyle \ln 2,}
and exponential and sine functions. It was proved in 1968 by mathematician and computer scientist Daniel Richardson of the University of Bath.
Specifically, the class of expressions for which the theorem holds is that generated by rational numbers, the number π, the number ln 2, the variable x, the operations of addition, subtraction, multiplication, composition, and the sin, exp, and abs functions | Richardson's theorem |
7,446 | Robinson's joint consistency theorem is an important theorem of mathematical logic. It is related to Craig interpolation and Beth definability.
The classical formulation of Robinson's joint consistency theorem is as follows:
Let
T
1
{\displaystyle T_{1}}
and
T
2
{\displaystyle T_{2}}
be first-order theories | Robinson's joint consistency theorem |
7,447 | The Cantor–Bernstein–Schroeder theorem of set theory has a counterpart for measurable spaces, sometimes called the Borel Schroeder–Bernstein theorem, since measurable spaces are also called Borel spaces. This theorem, whose proof is quite easy, is instrumental when proving that two measurable spaces are isomorphic. The general theory of standard Borel spaces contains very strong results about isomorphic measurable spaces, see Kuratowski's theorem | Schröder–Bernstein theorem for measurable spaces |
7,448 | In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B.
In terms of the cardinality of the two sets, this classically implies that if |A| ≤ |B| and |B| ≤ |A|, then |A| = |B|; that is, A and B are equipotent.
This is a useful feature in the ordering of cardinal numbers | Schröder–Bernstein theorem |
7,449 | This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
The binary operations of set union (
∪
{\displaystyle \cup }
) and intersection (
∩
{\displaystyle \cap }
) satisfy many identities | List of set identities and relations |
7,450 | In order theory, the Szpilrajn extension theorem (also called the order-extension principle), proved by Edward Szpilrajn in 1930, states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair becomes comparable. The theorem is one of many examples of the use of the axiom of choice in the form of Zorn's lemma to find a maximal set with certain properties | Szpilrajn extension theorem |
7,451 | In mathematical logic, Tarski's high school algebra problem was a question posed by Alfred Tarski. It asks whether there are identities involving addition, multiplication, and exponentiation over the positive integers that cannot be proved using eleven axioms about these operations that are taught in high-school-level mathematics. The question was solved in 1980 by Alex Wilkie, who showed that such unprovable identities do exist | Tarski's high school algebra problem |
7,452 | In mathematics, Tarski's theorem, proved by Alfred Tarski (1924), states that in ZF the theorem "For every infinite set
A
{\displaystyle A}
, there is a bijective map between the sets
A
{\displaystyle A}
and
A
×
A
{\displaystyle A\times A}
" implies the axiom of choice. The opposite direction was already known, thus the theorem and axiom of choice are equivalent.
Tarski told Jan Mycielski (2006) that when he tried to publish the theorem in Comptes Rendus de l'Académie des Sciences de Paris, Fréchet and Lebesgue refused to present it | Tarski's theorem about choice |
7,453 | Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that "arithmetical truth cannot be defined in arithmetic". The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system | Tarski's undefinability theorem |
7,454 | Tennenbaum's theorem, named for Stanley Tennenbaum who presented the theorem in 1959, is a result in mathematical logic that states that no countable nonstandard model of first-order Peano arithmetic (PA) can be recursive (Kaye 1991:153ff).
Recursive structures for PA
A structure
M
{\displaystyle M}
in the language of PA is recursive if there are recursive functions
⊕
{\displaystyle \oplus }
and
⊗
{\displaystyle \otimes }
from
N
×
N
{\displaystyle \mathbb {N} \times \mathbb {N} }
to
N
{\displaystyle \mathbb {N} }
, a recursive two-place relation <M on
N
{\displaystyle \mathbb {N} }
, and distinguished constants
n
0
,
n
1
{\displaystyle n_{0},n_{1}}
such that
(
N
,
⊕
,
⊗
,
<
M
,
n
0
,
n
1
)
≅
M
,
{\displaystyle (\mathbb {N} ,\oplus ,\otimes ,<_{M},n_{0},n_{1})\cong M,}
where
≅
{\displaystyle \cong }
indicates isomorphism and
N
{\displaystyle \mathbb {N} }
is the set of (standard) natural numbers. Because the isomorphism must be a bijection, every recursive model is countable | Tennenbaum's theorem |
7,455 | The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature | Ultraproduct |
7,456 | In mathematics, the von Neumann paradox, named after John von Neumann, is the idea that one can break a planar figure such as the unit square into sets of points and subject each set to an area-preserving affine transformation such that the result is two planar figures of the same size as the original. This was proved in 1929 by John von Neumann, assuming the axiom of choice. It is based on the earlier Banach–Tarski paradox, which is in turn based on the Hausdorff paradox | Von Neumann paradox |
7,457 | In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also Axiom of choice § Equivalents) | Well-ordering theorem |
7,458 | In mathematics, Wilkie's theorem is a result by Alex Wilkie about the theory of ordered fields with an exponential function, or equivalently about the geometric nature of exponential varieties.
Formulations
In terms of model theory, Wilkie's theorem deals with the language Lexp = (+, −, ·, <, 0, 1, ex), the language of ordered rings with an exponential function ex. Suppose φ(x1, | Wilkie's theorem |
7,459 | This is a list of notable economists, mathematicians, political scientists, and computer scientists whose work has added substantially to the field of game theory.
Derek Abbott - quantum game theory and Parrondo's games
Susanne Albers - algorithmic game theory and algorithm analysis
Kenneth Arrow - voting theory (Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel 1972)
Robert Aumann - equilibrium theory (Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel 2005)
Robert Axelrod - repeated Prisoner's Dilemma
Tamer Başar - dynamic game theory and application robust control of systems with uncertainty
Cristina Bicchieri - epistemology of game theory
Olga Bondareva - Bondareva–Shapley theorem
Steven Brams - cake cutting, fair division, theory of moves
Jennifer Tour Chayes - algorithmic game theory and auction algorithms
John Horton Conway - combinatorial game theory
Olivier Gossner - value of information, bounded rationality
William Hamilton - evolutionary biology
John Harsanyi - equilibrium theory (Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel 1994)
Monika Henzinger - algorithmic game theory and information retrieval
Naira Hovakimyan - differential games and adaptive control
Peter L. Hurd - evolution of aggressive behavior
Rufus Isaacs - differential games
Ehud Kalai - Kalai-Smorodinski bargaining solution, rational learning, strategic complexity
Anna Karlin - algorithmic game theory and online algorithms
Michael Kearns - algorithmic game theory and computational social science
Sarit Kraus - non-monotonic reasoning
Ehud Lehrer - Repeated games, approachability theory
John Maynard Smith - evolutionary biology
Oskar Morgenstern - social organization
John Forbes Nash - Nash equilibrium (Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel 1994)
John von Neumann - Minimax theorem, expected utility, social organization, arms race
Abraham Neyman - Stochastic games, Shapley value
J | List of game theorists |
7,460 | Susanne Albers is a German theoretical computer scientist and professor of computer science at the Department of Informatics of the Technical University of Munich. She is a recipient of the Otto Hahn Medal and the Leibniz Prize.
Education and career
Albers studied mathematics, computer science, and business administration in Osnabrück and received her PhD (Dr | Susanne Albers |
7,461 | Kenneth Joseph Arrow (23 August 1921 – 21 February 2017) was an American economist, mathematician, writer, and political theorist. He was the joint winner of the Nobel Memorial Prize in Economic Sciences with John Hicks in 1972.
In economics, he was a major figure in post-World War II neo-classical economic theory | Kenneth Arrow |
7,462 | Robert John Aumann (Hebrew name: ישראל אומן, Yisrael Aumann; born June 8, 1930) is an Israeli-American mathematician, and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel. He also holds a visiting position at Stony Brook University, and is one of the founding members of the Stony Brook Center for Game Theory | Robert Aumann |
7,463 | Robert Marshall Axelrod (born May 27, 1943) is an American political scientist. He is Professor of Political Science and Public Policy at the University of Michigan where he has been since 1974. He is best known for his interdisciplinary work on the evolution of cooperation | Robert Axelrod |
7,464 | John Francis Banzhaf III (; born July 2, 1940) is an American public interest lawyer, legal activist and law professor at George Washington University Law School. He is the founder of an antismoking advocacy group, Action on Smoking and Health. He is noted for his advocacy and use of lawsuits as a method to promote what he believes is the public interest | John Banzhaf |
7,465 | Mustafa Tamer Başar (born January 19, 1946) is a control and game theorist who is the Swanlund Endowed Chair and Center for Advanced Study Professor of Electrical and Computer Engineering at the University of Illinois at Urbana-Champaign, USA. He is also the Director of the Center for Advanced Study (since 2014).
Education
Tamer Başar received a B | Tamer Başar |
7,466 | Richard Ernest Bellman (August 26, 1920 – March 19, 1984) was an American applied mathematician, who introduced dynamic programming in 1953, and made important contributions in other fields of mathematics, such as biomathematics. He founded the leading biomathematical journal Mathematical Biosciences.
Biography
Bellman was born in 1920 in New York City to non-practising Jewish parents of Polish and Russian descent, Pearl (née Saffian) and John James Bellman, who ran a small grocery store on Bergen Street near Prospect Park, Brooklyn | Richard E. Bellman |
7,467 | Cristina Bicchieri (born 1950) is an Italian–American philosopher. She is the S. J | Cristina Bicchieri |
7,468 | Kenneth George "Ken" Binmore, (born 27 September 1940) is an English mathematician, economist, and game theorist, a Professor Emeritus of Economics at University College London (UCL) and a Visiting Emeritus Professor of Economics at the University of Bristol. As a founder of modern economic theory of bargaining (with Nash and Rubinstein), he made important contributions to the foundations of game theory, experimental economics, evolutionary game theory and analytical philosophy. He took up economics after holding the Chair of Mathematics at the London School of Economics | Kenneth Binmore |
7,469 | David Harold Blackwell (April 24, 1919 – July 8, 2010) was an American statistician and mathematician who made significant contributions to game theory, probability theory, information theory, and statistics. He is one of the eponyms of the Rao–Blackwell theorem. He was the first African American inducted into the National Academy of Sciences, the first African American tenured faculty member at the University of California, Berkeley, and the seventh African American to receive a Ph | David Blackwell |
7,470 | Olga Nikolaevna Bondareva (April 27, 1937 – December 9, 1991) was a distinguished Soviet mathematician and economist. She contributed to the fields of mathematical economics, especially game theory.
Bondareva is best known as one of the two independent discoverers of the Bondareva–Shapley theorem | Olga Bondareva |
7,471 | George Stephen Boolos (; 4 September 1940 – 27 May 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.
Life
Boolos was of Greek-Jewish descent. He graduated with an A | George Boolos |
7,472 | Steven J. Brams (born November 28, 1940 in Concord, New Hampshire) is an American game theorist and political scientist at the New York University Department of Politics. Brams is best known for using the techniques of game theory, public choice theory, and social choice theory to analyze voting systems and fair division | Steven Brams |
7,473 | Bruce Bueno de Mesquita (; born November 24, 1946) is a political scientist, professor at New York University, and senior fellow at Stanford University's Hoover Institution.
Biography
Bueno de Mesquita graduated from Stuyvesant High School in 1963, (along with Richard Axel and Alexander Rosenberg), earned his BA degree from Queens College, New York in 1967 and then his MA and PhD from the University of Michigan. He specializes in international relations, foreign policy, and nation building | Bruce Bueno de Mesquita |
7,474 | Colin Farrell Camerer (born December 4, 1959) is an American behavioral economist, and Robert Kirby Professor of Behavioral Finance and Economics at the California Institute of Technology (Caltech).
Background
A former child prodigy, Camerer received a B. A | Colin Camerer |
7,475 | Jennifer Tour Chayes is dean of the college of computing, data science, and society at the University of California, Berkeley. Before joining Berkeley, she was a technical fellow and managing director of Microsoft Research New England in Cambridge, Massachusetts, which she founded in 2008, and Microsoft Research New York City, which she founded in 2012. Chayes is best known for her work on phase transitions in discrete mathematics and computer science, structural and dynamical properties of self-engineered networks, and algorithmic game theory | Jennifer Tour Chayes |
7,476 | Graciela Chichilnisky (born 1944) is an Argentine American mathematical economist. She is a professor of economics at Columbia University and has expertise in climate change. She is also co-founder and former CEO of the company Global Thermostat | Graciela Chichilnisky |
7,477 | Vincent P. Crawford (born 1950) is an American economist. He is a senior research fellow at the University of Oxford, following his tenure as Drummond Professor of Political Economy from 2010 to 2020 | Vincent Crawford |
7,478 | Eddie Dekel (born September 28, 1958) is an Israeli-American economist. He is a professor at Northwestern University and Tel Aviv University. His fields of research include game theory and decision theory | Eddie Dekel |
7,479 | Melvin Dresher (born Dreszer; March 13, 1911 – June 4, 1992) was a Polish-born American mathematician, notable for developing, with Merrill Flood, the game theoretical model of cooperation and conflict known as the Prisoner's dilemma while at RAND in 1950 (Albert W. Tucker gave the game its prison-sentence interpretation, and thus the name by which it is known today).
Dresher came to the United States in 1923 | Melvin Dresher |
7,480 | Pradeep Dubey (born 9 January 1951) is an Indian game theorist. He is a Professor of Economics at the State University of New York, Stony Brook , and a member of the Stony Brook Center for Game Theory. He also holds a visiting position at Cowles Foundation, Yale University | Pradeep Dubey |
7,481 | Robert James Elliott (born 1940) is a British-Canadian mathematician, known for his contributions to control theory, game theory, stochastic processes and mathematical finance.
He was schooled at Swanwick Hall Grammar School in Swanwick, Derbyshire and studied mathematics in which he earn a B. A | Robert J. Elliott |
7,482 | Richard Arnold Epstein (born March 5, 1927 in Los Angeles, California), also known under the pseudonym E. P. Stein, is an American game theorist | Richard Arnold Epstein |
7,483 | Reginald Robin Farquharson (3 October 1930 – 1 April 1973) was an academic whose interest in mathematics and politics led him to work on game theory. He wrote an influential analysis of voting systems in his doctoral thesis, later published as Theory of Voting. Farquharson diagnosed himself as suffering from bipolar disorder (manic depression), and episodes of mania made it difficult for him to obtain a permanent university position and also resulted in him losing commercial employment | Robin Farquharson |
7,484 | Merrill Meeks Flood (1908 – 1991) was an American mathematician, notable for developing, with Melvin Dresher, the basis of the game theoretical Prisoner's dilemma model of cooperation and conflict while being at RAND in 1950 (Albert W. Tucker gave the game its prison-sentence interpretation, and thus the name by which it is known today).
Biography
Flood received an MA in mathematics at the University of Nebraska, and a PhD at Princeton University in 1935 under the supervision of Joseph Wedderburn, for the dissertation Division by Non-singular Matric Polynomials | Merrill M. Flood |
7,485 | Françoise Forges (born 3 July 1958) is a Belgian and French economist known for her work in game theory. She is professor of economics at Paris Dauphine University.
Education and career
Forges was born on 3 July 1958 in Brussels, but is a French citizen | Françoise Forges |
7,486 | Drew Fudenberg (born March 2, 1957) is a Professor of Economics at MIT. His extensive research spans many aspects of game theory, including equilibrium theory, learning in games, evolutionary game theory, and many applications to other fields. Fudenberg was also one of the first to apply game theoretic analysis in industrial organization, bargaining theory, and contract theory | Drew Fudenberg |
7,487 | Shmuel Gal (Hebrew: שמואל גל, born 1940) is a mathematician and professor of statistics at the University of Haifa in Israel.
He devised the Gal's accurate tables method for the computer evaluation of elementary functions. With Zvi Yehudai he developed in 1993 a new algorithm for sorting which is used by IBM | Shmuel Gal |
7,488 | David Gale (December 13, 1921 – March 7, 2008) was an American mathematician and economist. He was a professor emeritus at the University of California, Berkeley, affiliated with the departments of mathematics, economics, and industrial engineering and operations research. He has contributed to the fields of mathematical economics, game theory, and convex analysis | David Gale |
7,489 | Allan Fletcher Gibbard (born 1942) is the Richard B. Brandt Distinguished University Professor of Philosophy Emeritus at the University of Michigan, Ann Arbor. Gibbard has made major contributions to contemporary ethical theory, in particular metaethics, where he has developed a contemporary version of non-cognitivism | Allan Gibbard |
7,490 | Donald Bruce Gillies (October 15, 1928 – July 17, 1975) was a Canadian computer scientist and mathematician who worked in the fields of computer design, game theory, and minicomputer programming environments.
Early life and education
Donald B. Gillies was born in Toronto, Ontario, Canada, to John Zachariah Gillies (a Canadian) and Anne Isabelle Douglas MacQueen (an American) | Donald B. Gillies |
7,491 | Herbert Gintis (February 11, 1940 – January 5, 2023) was an American economist, behavioral scientist, and educator known for his theoretical contributions to sociobiology, especially altruism, cooperation, epistemic game theory, gene-culture coevolution, efficiency wages, strong reciprocity, and human capital theory. Throughout his career, he worked extensively with economist Samuel Bowles. Their landmark book, Schooling in Capitalist America, had multiple editions in five languages since it was first published in 1976 | Herbert Gintis |
7,492 | Patrick Grim is an American philosopher. He has published on epistemic questions in philosophy of religion, as well as topics in philosophy of science, philosophy of logic, computational philosophy, and agent-based modeling. He is author, co-author or editor of seven books in philosophical logic, philosophy of mind, philosophy of science and computational philosophy | Patrick Grim |
7,493 | Marina Halac (born November 17, 1979) is a professor of economics at Yale University. She is also an associate editor of Econometrica and a member of the editorial board of the American Economic Review. She was the 2016 recipient of the Elaine Bennett Research Prize, which is awarded biennially by the American Economic Association to recognize outstanding research by a woman | Marina Halac |
7,494 | Mahbub ul-Haq (Urdu: محبوب الحق; (1934-02-24)24 February 1934 – (1998-07-16)16 July 1998) was a Pakistani economist, international development theorist, and politician who served as the Minister of Finance of Pakistan from 10 April 1985 to 28 January 1986, and again from June to December 1988 as a caretaker. Regarded as one of the greatest economists of his time, Haq devised the Human Development Index, widely used to gauge the development of nations. After graduating with a degree in economics from the Government College University in Lahore, he won a scholarship to the University of Cambridge in England, where he obtained a second higher degree in the same field | Mahbub ul Haq |
7,495 | John Charles Harsanyi (Hungarian: Harsányi János Károly; May 29, 1920 – August 9, 2000) was a Hungarian-American economist and the recipient of the Nobel Memorial Prize in Economic Sciences in 1994.
He is best known for his contributions to the study of game theory and its application to economics, specifically for his developing the highly innovative analysis of games of incomplete information, so-called Bayesian games. He also made important contributions to the use of game theory and economic reasoning in political and moral philosophy (specifically utilitarian ethics) as well as contributing to the study of equilibrium selection | John Harsanyi |
7,496 | Sergiu Hart (Hebrew: סרג'יו הרט) (born 1949) is an Israeli mathematician and economist. He is the Chairperson of the Humanities Division of the Israel Academy of Sciences and Humanities, and the past President of the Game Theory Society (2008–2010). He also is emeritus professor of mathematics at the Kusiel-Vorreuter University, and the emeritus professor of economics at the Center for the Study of Rationality at the Hebrew University of Jerusalem in Israel | Sergiu Hart |
7,497 | Monika Henzinger (born as Monika Rauch, 17 April 1966 in Weiden in der Oberpfalz) is a German computer scientist, and is a former director of research at Google. She is currently a professor at the University of Vienna. Her expertise is mainly on algorithms with a focus on data structures, algorithmic game theory, information retrieval, search algorithms and Web data mining | Monika Henzinger |
7,498 | Kaarlo Jaakko Juhani Hintikka (12 January 1929 – 12 August 2015) was a Finnish philosopher and logician.
Life and career
Hintikka was born in Helsingin maalaiskunta (now Vantaa).
In 1953, he received his doctorate from the University of Helsinki for a thesis entitled Distributive Normal Forms in the Calculus of Predicates | Jaakko Hintikka |
7,499 | James Martin Hollis (14 March 1938 – 27 February 1998) was an English rationalist philosopher.
Writing for The Independent, Tim O'Hagan, an Emeritus Professor at the University of East Anglia argued that central to Hollis's rationalism was "the epistemological unity of mankind", the view that "some beliefs are universal . | Martin Hollis (philosopher) |
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