Communication
Free
Faster access than browser!

# S (set theory)

S is an axiomatic set theory set out by George Boolos in his article, Boolos (1989). [1]

## Abstract and concrete

Abstract and concrete are classifications that denote whether a term describes an object with a physical referent or one with no physical referents.

## Axiom of choice

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.

## Axiom of empty set

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.

## Axiom of extensionality

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.

## Axiom of infinity

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.

## Axiom of pairing

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.

## Axiom schema of replacement

In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZFC) that asserts that the image of any set under any definable mapping is also a set.

## Axiom schema of specification

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 naïvely 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 derivable from the theorem that there is no greatest cardinal number, so that the collection of "infinite sizes" is itself infinite.

## Category theory

Category theory formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows (also called morphisms).

## Class (set theory)

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.

## Domain of discourse

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.

## Element (mathematics)

In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.

## Empty set

In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

## Extensionality

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

First-order logic is a formal system used in mathematics, philosophy, linguistics, and computer science.

## George Boolos

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.

## Gottlob Frege

Friedrich Ludwig Gottlob Frege (8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician.

## Hierarchy (mathematics)

In mathematics, a hierarchy is a set-theoretical object, consisting of a preorder defined on a set.

## Hume's principle

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.

## Identity (mathematics)

In mathematics an identity is an equality relation A.

## If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

## Inaccessible cardinal

In set theory, an uncountable regular cardinal number is called weakly inaccessible if it is a weak limit cardinal, and strongly inaccessible, or just inaccessible, if it is a strong limit cardinal.

## List of first-order theories

In mathematical logic, a first-order theory is given by a set of axioms in some language.

## Mathematical object

A mathematical object is an abstract object arising in philosophy of mathematics and mathematics itself.

## Naive set theory

Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics.

## Ontology

Ontology is the philosophical study of the nature of being, becoming, existence, or reality, as well as the basic categories of being and their relations.

## Ordinal number

In set theory, an ordinal number, or ordinal, is the order type of a well-ordered set.

A paradox is a statement that apparently contradicts itself and yet might be true (or wrong at the same time).

## Predicate (mathematical logic)

In mathematics, 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.

## Primitive notion

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 naive set theory created by Georg Cantor led to a contradiction.

## Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

## Set theory

Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects.

## Tarski–Grothendieck set theory

Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory.

## Transfinite number

Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite.

## Transitive relation

In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c. Transitivity is a key property of both partial order relations and equivalence relations.

## Urelement

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.

## Von Neumann universe

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.

## Well-order

In mathematics, a well-order relation (or well-ordering) 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

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

Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory.

## Zermelo–Fraenkel set theory

In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets free of paradoxes such as Russell's paradox.

## References

Hey! We are on Facebook now! »