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...
 Foundations of mathematics - Unsolved Problems
 Foundations of mathematics - Math World
 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...