Foundations of mathematics
Foundations of mathematics is the study of the logical and philosophical basis of mathematics, or, in a broader sense, the mathematical investigation of the consequences of what are at bottom philosop...
Axiom
An axiom or postulate is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy.The word comes from the Gree...
Metatheorem
In logic, a metatheorem is a statement about a formal system proven in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may referen...
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of ...
Theorem - Wikipedia
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be trac...
Category theory
Category theory is used to formalize mathematical structure and its concepts as a collection of objects and arrows (also called morphisms). A category has two basic properties: the ability to compose ...
Set theory
Set theory is the branch of mathematical logic that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects...
Intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activ...
Logicism
Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to log...
Zermelo-Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems th...
Gödel's completeness theorem
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. It makes a close li...
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, pro...
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provi...
Philosophy of mathematics - Wikipedia
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
"The Unreasonable Effectiveness of Mathematics in the Natural Sciences" is the title of an article published in 1960 by the physicist Eugene Wigner. In the paper, Wigner observed that the mathematical...
Mereotopology
In formal ontology, a branch of metaphysics, and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes,...
Probabilistic proofs of non-probabilistic theorems
Probability theory routinely uses results from other fields of mathematics (mostly, analysis). The opposite cases, collected below, are relatively rare; however, probability theory is used systematica...
Tarski's axioms
Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called "elementary," that is formulable in first-order logic with identity, and requiring no...
Tarski's axioms - Wikipedia
Kummer's theorem
In mathematics, Kummer's theorem on binomial coeffients gives the highest power of a prime number p dividing a binomial coefficient. In particular, it asserts that given integers n ≥ m ...
Mathematical intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activ...
Axioms of set theory
Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often...
Huzita–Hatori axioms
The Huzita–Hatori axioms or Huzita–Justin axioms are a set of rules related to the mathematical principles of paper folding, describing the operations that can be made when folding a piece of paper. T...
Huzita–Hatori axioms - Wikipedia
Bauer-Fike theorem
In mathematics, the Bauer–Fike theorem is a standard result in the perturbation theory of the eigenvalue of a complex-valued diagonalizable matrix. In its substance, it states an absolute upper bound ...
Recursive set
In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithm which terminates after a finite amount of time and correctly decides whether or ...
Fuzzy set
In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced by Lotfi A. Zadeh and Dieter Klaua in 1965 as an extension of the classical notion of set. At...
St. Petersburg paradox
The St. Petersburg lottery or St. Petersburg paradox is a paradox related to probability and decision theory in economics. It is based on a particular (theoretical) lottery game that leads to a random...
Axiom of dependent choice
In mathematics, the axiom of dependent choice, denoted DC, is a weak form of the axiom of choice (AC) that is still sufficient to develop most of real analysis.
The axiom can be stated as follows:...
Kuratowski closure axioms
In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly...
Cantor–Dedekind axiom
In mathematical logic, the phrase Cantor–Dedekind axiom has been used to describe the thesis that the real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom s...
Borel determinacy theorem
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 strate...