In the axiomatization of set theory known as Zermelo-Fraenkel set theory, the existence of the empty set is assured by the axiom of empty set. The uniqueness of the empty set follows from the axiom of extensionality. Any axiom that states the existence of any set will imply the axiom of empty set, using the axiom schema of separation. For example, if A is a set then the axiom schema of separation allows the construction of the set B = {x in A | x ≠ See also:

Empty set, Empty set - Notation, Empty set - Properties, Empty set - Common problems, Empty set - Axiomatic set theory, Empty set - Does it exist or is it necessary?, Empty set - Operations on the empty set, Empty set - Bounds, Empty set - The empty set and zero, Empty set - Category theory

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Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties. Formal versions of set theory also have a foundational role to play as specifying a theoretical ideal of mathematical rig ...

Including:

• Axiomatic set theory - The origins of rigorous set theory
• Axiomatic set theory - Axioms for set theory
• Axiomatic set theory - Independence in ZFC
• Axiomatic set theory - Set theory ZFC foundations for mathematics
• Axiomatic set theory - Well-foundedness and hypersets
• Axiomatic set theory - Objections to set theory

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The important idea of Cantor's, which got set theory going as a new field of study, was to define two sets A and B to have the same number of members (the same cardinality) when there is a way of pairing off members of A exhaustively with members of B. Then the set N of natural numbers has the same cardinality as the set Q of rational numbers (they are both said to be countably infinite), even though N is a proper subset of Q. On the other hand, the set R of real numbers d ...

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Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory

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The Zermelo-Fraenkel axioms of set theory together with the axiom of choice are the standard axioms of axiomatic set theory. All of ordinary mathematics can be based on this axiom system. The Zermelo-Fraenkel axioms without the axiom of choice are usually denoted by ZF. The ZF axioms together with the axiom of choice (AC) are denoted ZFC. The axioms are the result of work by Thoralf Skolem in 1922, based on earlier work by Abraham Fraenkel in the same year, which was based on the axi ...

Including:

• Zermelo-Fraenkel set theory - The Axioms

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In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. Not all epistemologists agree that any axioms, understood in that sense, exist. In mathematics, an axiom is not necessarily a self-evident truth but rather, a formal logical expression used in a deduction to yield further results. Mathematics distinguishes two types of axioms: logical axioms and non-logical axioms. Axiom - Etymology. The word axiomIncluding:

• Axiom - Etymology
• Axiom - Mathematics
• Axiom - Logical axioms
• Axiom - Non-logical axioms
• Axiom - Role in mathematical logic
• Axiom - Further discussion

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In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical axioms and non-logical axioms. Axiom - Logical axioms. These are certain formulas in a language that are universally valid, that is, formulas that are satisfied by every structure under every variable assignment function . More colloquially, these are statements that are true in any possible universe, under any possible interpretation and with any assignment of values. Usually one takes as logical axioms some minimal set of tautologies that is sufficient for proving all ...

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Axiom, Axiom - Etymology, Axiom - Mathematics, Axiom - Logical axioms, Axiom - Non-logical axioms, Axiom - Role in mathematical logic, Axiom - Further discussion

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The axioms of ZFC are: Axiom of extensionality: Two sets are the same if and only if they have the same elements. Axiom of empty set: There is a set with no elements. We will also use {} to denote this empty set. Axiom of pairing: If x, y are sets, then there exists a set containing x and y as its only elements, which we denote by {x,y} or {x} ∪ {y}. < ...

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Zermelo-Fraenkel set theory, Zermelo-Fraenkel set theory - The Axioms

Read more here: » Zermelo-Fraenkel set theory: Encyclopedia II - Zermelo-Fraenkel set theory - The Axioms

Given two sets A and B, we may construct their union. This is the set consisting of all objects which are elements of A or of B or of both (see axiom of union). It is denoted by A ∪ B. The intersection of A and B is the set of all objects which are both in A and in B. It is denoted by A ∩ B. Finally, the relative complement of B relative to A, also known as the set theoretic differenc ...

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Naive set theory, Naive set theory - Introduction, Naive set theory - Sets membership and equality, Naive set theory - Specifying sets, Naive set theory - Subsets, Naive set theory - Universal sets and absolute complements, Naive set theory - Unions intersections and relative complements, Naive set theory - Ordered pairs and Cartesian products, Naive set theory - Some important sets, Naive set theory - Paradoxes, Naive set theory - Footnote

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In this section we usually consider NFU, which is known to be consistent, and discuss the effect of various "strong axioms of infinity". One can add such axioms as "there is an inaccessible cardinal" to NFU, but it is more natural to consider principles appropriate to this theory, usually assertions about cantorian and strongly cantorian sets: these are ways to strengthen the theory in its own terms, and they have the effect of causing large cardinals of the usual sorts to appear. NFU + Infinity + Choice, our usual base theory, has the same strength as TST + Infinity or Zermelo set t ...

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New Foundations, New Foundations - The type theory TST, New Foundations - Definition of New Foundations; stratification, New Foundations - Large sets appear, New Foundations - The consistency problem and related partial results, New Foundations - Resolving the paradoxes in NFU, New Foundations - Models of NFU, New Foundations - Self-sufficiency of mathematical foundations in NFU, New Foundations - Facts about the automorphism j, New Foundations - Strong Principles

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The outstanding problem with this theory is the problem as to whether it is consistent. It is known that NF disproves Choice, and so proves Infinity (Specker, 1953). But it is also known (Jensen, 1969) that the minor(?) modification of allowing urelements (objects which have no elements but which are distinct from the empty set and from one another) produces a theory NFU which is known to be consistent (weaker than arithmetic) and consistent with Infinity and Choice as well. (This corresponds to a type theory TSTU in which po ...

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New Foundations, New Foundations - The type theory TST, New Foundations - Definition of New Foundations; stratification, New Foundations - Large sets appear, New Foundations - The consistency problem and related partial results, New Foundations - Resolving the paradoxes in NFU, New Foundations - Models of NFU, New Foundations - Self-sufficiency of mathematical foundations in NFU, New Foundations - Facts about the automorphism j, New Foundations - Strong Principles

Read more here: » New Foundations: Encyclopedia II - New Foundations - The consistency problem and related partial results

New Foundations (NF) is obtained from this theory by abandoning the distinctions of type. The axioms of this theory are extensionality (two objects with the same elements are the same object) and all those instances of comprehension obtained by dropping type indices (without introducing new identifications between variables) from an instance of comprehension of the streamlined type theory. It might seem that the resulting comprehension scheme would be the inconsistent scheme of naive set theory, but this is not the case: for example, the imp ...

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New Foundations, New Foundations - The type theory TST, New Foundations - Definition of New Foundations; stratification, New Foundations - Large sets appear, New Foundations - The consistency problem and related partial results, New Foundations - Resolving the paradoxes in NFU, New Foundations - Models of NFU, New Foundations - Self-sufficiency of mathematical foundations in NFU, New Foundations - Facts about the automorphism j, New Foundations - Strong Principles

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Theory has a number of distinct meanings in different fields of knowledge, depending on the context and their methodologies. Theory - Etymology. The word ‘theory’ derives from the Greek ‘theorein’, which means ‘to look at’. According to some sources, it was used frequently in terms of ‘looking at’ a theatre stage, which may explain why sometimes the word ‘theory’ is used as something provisional or not completely resembling real. The term ‘theoria’ (a noun) was already used by ...

Including:

• Theory - Etymology
• Theory - Science
• Theory - Models
• Theory - Types of theories
• Theory - Further explanation of a scientific theory
• Theory - Characteristics
• Theory - Mathematics
• Theory - Other fields
• Theory - List of famous theories
• Theory - Reference

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Here we discuss a fairly simple method for producing models of NFU in bulk. Using well-known techniques of model theory one can construct a nonstandard model of Zermelo set theory (nothing nearly as strong as full ZFC is needed for the basic technique) on which there is an external automorphism j (not a set of the model) which moves a rank Vα of the cumulative hierarchy of sets. We may suppose without loss of generality that j(α) < α. We talk abou ...

See also:

New Foundations, New Foundations - The type theory TST, New Foundations - Definition of New Foundations; stratification, New Foundations - Large sets appear, New Foundations - The consistency problem and related partial results, New Foundations - Resolving the paradoxes in NFU, New Foundations - Models of NFU, New Foundations - Self-sufficiency of mathematical foundations in NFU, New Foundations - Facts about the automorphism j, New Foundations - Strong Principles

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We review the paradoxes of set theory and their resolution in NF, noting that although the consistency of NF remains an open question, these solutions to the paradoxes can be seen to work in NFU, which is known to be consistent. Relations and functions are defined in NF or NFU as sets of ordered pairs in the usual way. The Kuratowski ordered pair can be defined in this theory, but has the disadvantage that it is two types higher than its projections in TST, which means that a function is three types higher than the elements of its dom ...

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New Foundations, New Foundations - The type theory TST, New Foundations - Definition of New Foundations; stratification, New Foundations - Large sets appear, New Foundations - The consistency problem and related partial results, New Foundations - Resolving the paradoxes in NFU, New Foundations - Models of NFU, New Foundations - Self-sufficiency of mathematical foundations in NFU, New Foundations - Facts about the automorphism j, New Foundations - Strong Principles

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In naive set theory, a set is described as a well-defined collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even integers. As can be seen from this example, sets are allowed to have an infinite number of elements. If x is a member of A, then it is also said that x belongs to A, or that x is in A. In this case, we write x&# ...

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Naive set theory, Naive set theory - Introduction, Naive set theory - Sets membership and equality, Naive set theory - Specifying sets, Naive set theory - Subsets, Naive set theory - Universal sets and absolute complements, Naive set theory - Unions intersections and relative complements, Naive set theory - Ordered pairs and Cartesian products, Naive set theory - Some important sets, Naive set theory - Paradoxes, Naive set theory - Footnote

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Given two sets A and B we say that A is a subset of B if every element of A is also an element of B. Notice that in particular, B is a subset of itself; a subset of B that isn't equal to B is called a proper subset. If A is a subset of B, then one can also say that B is a superset of A, that A is contained in B, or that B contains A. In symbols, A ⊆ B means ...

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Naive set theory, Naive set theory - Introduction, Naive set theory - Sets membership and equality, Naive set theory - Specifying sets, Naive set theory - Subsets, Naive set theory - Universal sets and absolute complements, Naive set theory - Unions intersections and relative complements, Naive set theory - Ordered pairs and Cartesian products, Naive set theory - Some important sets, Naive set theory - Paradoxes, Naive set theory - Footnote

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The streamlined version of the theory of types (which we call TST) has a linear hierarchy of types: type 0 consists of individuals otherwise undescribed, while for each (meta-) natural number n, type n+1 objects are understood to be sets of type n objects. The primitive predicates of this theory are equality and membership. Typing rules for well-formed atomic sentences are succinctly summed up in the formulas xn = yn and (notation to be impro ...

See also:

New Foundations, New Foundations - The type theory TST, New Foundations - Definition of New Foundations; stratification, New Foundations - Large sets appear, New Foundations - The consistency problem and related partial results, New Foundations - Resolving the paradoxes in NFU, New Foundations - Models of NFU, New Foundations - Self-sufficiency of mathematical foundations in NFU, New Foundations - Facts about the automorphism j, New Foundations - Strong Principles

Read more here: » New Foundations: Encyclopedia II - New Foundations - The type theory TST

The simplest way to describe a set is to list its elements between curly braces. Thus {1,2} denotes the set whose only elements are 1 and 2. (See axiom of pairing.) Note the following points: Order of elements is immaterial; for example, {1,2} = {2,1}. Repetition (multiplicity) of elements is irrelevant; for example, {1,2,2} = {1,1,1,2} = {1,2}. (These are consequences of the definition of equality in the previous section.) This notation can be informally abused by saying something like {dogs} to indicate the set of all dogs, but this example would usually be read by mathematicians as ...

See also:

Naive set theory, Naive set theory - Introduction, Naive set theory - Sets membership and equality, Naive set theory - Specifying sets, Naive set theory - Subsets, Naive set theory - Universal sets and absolute complements, Naive set theory - Unions intersections and relative complements, Naive set theory - Ordered pairs and Cartesian products, Naive set theory - Some important sets, Naive set theory - Paradoxes, Naive set theory - Footnote

Read more here: » Naive set theory: Encyclopedia II - Naive set theory - Specifying sets

Naive set theory was created at the end of the 19th century by Georg Cantor in order to allow mathematicians to work with infinite sets consistently. As it turned out, assuming that one could perform any operations on sets without restriction led to paradoxes such as Russell's paradox. In response, axiomatic set theory was developed to determine precisely what operations were allowed and when. Today, when mathematicians talk about "set theory" as a field, they usually mean axiomatic set theory, but when they talk about set theory as a mere tool to be applied to other mathema ...

See also:

Naive set theory, Naive set theory - Introduction, Naive set theory - Sets membership and equality, Naive set theory - Specifying sets, Naive set theory - Subsets, Naive set theory - Universal sets and absolute complements, Naive set theory - Unions intersections and relative complements, Naive set theory - Ordered pairs and Cartesian products, Naive set theory - Some important sets, Naive set theory - Paradoxes, Naive set theory - Footnote

Read more here: » Naive set theory: Encyclopedia II - Naive set theory - Introduction

In certain contexts we may consider all sets under consideration as being subsets of some given universal set. For instance, if we are investigating properties of the real numbers R (and subsets of R), then we may take R as our universal set. It is important to realise that a universal set is only temporarily defined by the context; there is no such thing as a "universal" universal set, "the set of everything" (see Paradoxes below). Given a universal set U and a subset A of U, we may de ...

See also:

Naive set theory, Naive set theory - Introduction, Naive set theory - Sets membership and equality, Naive set theory - Specifying sets, Naive set theory - Subsets, Naive set theory - Universal sets and absolute complements, Naive set theory - Unions intersections and relative complements, Naive set theory - Ordered pairs and Cartesian products, Naive set theory - Some important sets, Naive set theory - Paradoxes, Naive set theory - Footnote

Read more here: » Naive set theory: Encyclopedia II - Naive set theory - Universal sets and absolute complements