534 articles
Axiom of determinacy
The axiom of determinacy, introduced by Mycielski and Steinhaus in 1962 as an alternative to the axiom of choice, asserts that every infinite two-player game is determined.
Axiom of empty set
The axiom of empty set, part of Zermelo's 1908 axioms, asserts that a set with no elements exists; with extensionality it is unique and serves as zero in the von Neumann natural numbers.
Axiom of extensionality
The axiom of extensionality is the axiom of set theory asserting that two sets are equal exactly when they share the same elements, first axiomatized by Ernst Zermelo in 1908.
Axiom of global choice
The axiom of global choice, or global choice, strengthens the axiom of choice by asserting one class function that selects an element from every nonempty set.
Axiom of infinity
The axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory, asserting the existence of an inductive set and hence of infinite sets like the natural numbers.
Axiom of pairing
The axiom of pairing, also called the axiom of the unordered pair, is a Zermelo–Fraenkel axiom: for any two sets x and y, the set {x, y} exists.
Axiom of power set
The axiom of power set, part of Zermelo–Fraenkel set theory, asserts that every set has a power set containing all its subsets, grounding Cantor's theorem.
Axiom of projective determinacy
The axiom of projective determinacy (PD) is the set-theoretic statement that every projective set of reals is determined; it is consistent with, but independent of, ZFC.
Axiom of regularity
The axiom of regularity, also called the axiom of foundation, is a Zermelo–Fraenkel set theory axiom requiring every non-empty set to have an element disjoint from it.
Axiom of union
The axiom of union, or sum-set axiom, is one of the Zermelo–Fraenkel axioms: for any set x, there is a set containing exactly the members of x's members.
Axiom schema of replacement
The axiom schema of replacement, also called the axiom of replacement, states that in Zermelo–Fraenkel set theory the image of a set under a definable function is a set.
Axiom schema of specification
The axiom schema of specification, also called the axiom schema of separation or subset axiom, is a Zermelo–Fraenkel axiom, introduced by Zermelo in 1908 to block Russell's paradox.
Axiomatic method
The axiomatic method builds a body of knowledge from a small set of axioms, deriving every other claim by explicit rules of inference, from Euclid's Elements onward.
Axiomatic system
An axiomatic system, or axiom system, is a body of knowledge organized as primitive terms, axioms accepted without proof, and rules of inference that derive further statements called theorems.
Baire property
The Baire property, also called the property of Baire, is a regularity property of subsets of a topological space: the set differs from an open set only by a meager set.
Baire space (set theory)
Baire space in set theory is the set of all infinite sequences of natural numbers with the product topology; it is homeomorphic to the irrational real numbers.
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem of set-theoretic geometry, proved by Stefan Banach and Alfred Tarski in 1924, stating that a solid ball can be reassembled into two identical balls.
Barber paradox
The barber paradox is a logic puzzle asking whether a barber who shaves all and only men who do not shave themselves shaves himself, since no such barber can exist.
Begriffsschrift
Begriffsschrift (German for "concept-script") is a formal logical system and two-dimensional notation introduced by Gottlob Frege in his 1879 booklet, containing the first quantifier in formal logic and the first logical system with relations.
Binary decision diagram
A binary decision diagram (BDD) is a rooted directed acyclic graph that represents a Boolean function compactly; Bryant's 1986 ordered form made it canonical and useful in formal verification.
Binary relation
A binary relation, also known as a dyadic or two-place relation, is a set of ordered pairs recording which elements of one set relate to which elements of another.
Biological network inference
Biological network inference is the reconstruction of a biological system's interaction structure, usually a gene regulatory network, from high-throughput data using methods like Bayesian networks and deep learning.
Bisimulation
Bisimulation (bisimilarity) is a relation between the states of two transition systems guaranteeing behavioral indistinguishability, introduced for automata by David Park in 1981 and extended by Robin Milner to communicating processes.
Boolean function
A Boolean function, also called a switching function, is a function whose inputs and output are 0 or 1, basic to propositional logic, digital circuit design, and cryptography.
Boolean satisfiability problem
The Boolean satisfiability problem (SAT) asks whether a Boolean formula has an assignment of true and false values that makes it true, and was the first problem proved NP-complete.
Boolean-valued model
A Boolean-valued model is a mathematical structure in which statements take truth values in a complete Boolean algebra, developed by Scott and Solovay as a presentation of forcing.
Borel determinacy theorem
The Borel determinacy theorem, or Borel determinacy, states that every infinite two-player game whose Borel payoff set gives one player a winning strategy, proved by Donald A. Martin in 1975.
Borel set
A Borel set, or Borel subset, is a subset of a topological space belonging to the Borel σ-algebra, the smallest σ-algebra containing every open set.
Bounded arithmetic
Bounded arithmetic is a family of weak subsystems of Peano arithmetic, started by Rohit Parikh in 1971, whose provably total functions characterize complexity classes like polynomial time.
Bulletin of Symbolic Logic
The Bulletin of Symbolic Logic is a quarterly peer-reviewed journal founded in 1995 by the Association for Symbolic Logic, publishing accessible accounts of developments in logic.