43 relations: Abstract and concrete, Axiom of choice, Axiom of empty set, Axiom of extensionality, Axiom of infinity, Axiom of pairing, Axiom schema of replacement, Axiom schema of specification, Burali-Forti paradox, Cantor's paradox, Category theory, Class (set theory), Domain of discourse, Element (mathematics), Empty set, Extensionality, First-order logic, George Boolos, Gottlob Frege, Hierarchy (mathematics), Hume's principle, Identity (mathematics), If and only if, Inaccessible cardinal, Mathematical object, Naive set theory, Ontology, Ordinal number, Paradox, Predicate (mathematical logic), Primitive notion, Russell's paradox, Set (mathematics), Set theory, Tarski–Grothendieck set theory, Transfinite number, Transitive relation, Urelement, Von Neumann universe, Well-order, Z, Zermelo set theory, Zermelo–Fraenkel set theory.
Abstract and concrete are classifications that denote whether a term describes an object with a physical referent or one with no physical referents.
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.
In axiomatic set theory, the axiom of empty set is an axiom of Kripke–Platek set theory and the variant of general set theory that Burgess (2005) calls "ST," and a demonstrable truth in Zermelo set theory and Zermelo–Fraenkel set theory, with or without the axiom of choice.
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory.
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory.
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory.
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set.
In many popular versions of axiomatic set theory the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema.
In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction.
In set theory, Cantor's paradox is a statement derivable from the theorem that there is no greatest cardinal number, so that the collection of "infinite sizes" is itself infinite.
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).
In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share.
In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range.
In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.
In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.
In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties.
First-order logic—also known as first-order predicate calculus and predicate logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.
George Stephen Boolos (September 4, 1940 – May 27, 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.
Friedrich Ludwig Gottlob Frege (8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician.
In mathematics, a hierarchy is a set-theoretical object, consisting of a preorder defined on a set.
Hume's principle or HP—the terms were coined by George Boolos—says that the number of Fs is equal to the number of Gs if and only if there is a one-to-one correspondence (a bijection) between the Fs and the Gs.
In mathematics an identity is an equality relation A.
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.
In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic.
A mathematical object is an abstract object arising in mathematics.
Naïve set theory is any of several theories of sets used in the discussion of the foundations of mathematics.
Ontology (introduced in 1606) is the philosophical study of the nature of being, becoming, existence, or reality, as well as the basic categories of being and their relations.
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.
A paradox is a statement that, despite apparently sound reasoning from true premises, leads to an apparently self-contradictory or logically unacceptable conclusion.
In mathematical logic, a predicate is commonly understood to be a Boolean-valued function P: X→, called the predicate on X. However, predicates have many different uses and interpretations in mathematics and logic, and their precise definition, meaning and use will vary from theory to theory.
In mathematics, logic, and formal systems, a primitive notion is an undefined concept.
In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naïve set theory created by Georg Cantor led to a contradiction.
In mathematics, a set is a collection of distinct objects, considered as an object in its own right.
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.
Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory.
Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite.
In mathematics, a binary relation over a set is transitive if whenever an element is related to an element and is related to an element then is also related to.
In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ur-, 'primordial') is an object (concrete or abstract) that is not a set, but that may be an element of a set.
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets.
In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.
Z (named zed or zee "Z", Oxford English Dictionary, 2nd edition (1989); Merriam-Webster's Third New International Dictionary of the English Language, Unabridged (1993); "zee", op. cit.) is the 26th and final letter of the modern English alphabet and the ISO basic Latin alphabet.
Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory.
In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox.