Axiomatic set theory

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.

Set theory (ZFC) foundations for mathematics

From these initial axioms for sets one can construct all other mathematical concepts and objects: number - discrete and continuous, order, relation, function, etc.

Related Topics:
Number - Discrete - Continuous - Order - Relation - Function

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For example, whilst the elements of a set have no intrinsic ordering it is possible to construct models of ordered lists. The essential step is to be able to model the ordered pair ( a, b ) which represents the pairing of two objects in this order. The defining property of an ordered pair is that ( a, b ) = ( c, d ) if and only if a = c and b = d. The approach is basically to specify the two elements and additionally note which one is the first using the construction:

Related Topics:
Ordered lists - Ordered pair

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( a, b ) = { { a, b }, { a } }.

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Ordered lists of greater length can be constructed inductively:

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( a, b, c ) = ( ( a, b ), c )

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( a, b, c, d ) = ( ( a, b, c ), d )

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...

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For another example, there is a minimalist construction for the natural numbers, principally drawing on the axiom of infinity, due to von Neumann. We require to produce an infinite sequence of distinct sets with a 'successor' relation as a model for the Peano Axioms. This provides a canonical representation for the number N as being a particular choice of set containing precisely N distinct elements.

Related Topics:
Von Neumann - Peano Axioms

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We proceed inductively:

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0 = {}

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1 = { 0 } = { {} }

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2 = { 0, 1 } = { {}, { {} } }

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3 = { 0, 1, 2 } = { {}, { {} }, { {}, { {} } } }

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...

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At each stage we construct a new set with N elements as being the set containing the (already defined) elements 0, 1, 2, ..., N - 1. More formally, at each step the successor of N is N  ∪  { N }. Remarkably this produces a suitable model for the entire collection of natural numbers - from the barest of materials.

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Since relations, and most specifically functions, are defined to be sets of ordered pairs, and there are well-known constructions progressively building up the integers, rational, real and complex numbers from sets of the natural numbers we are able to model essentially all of the usual infrastructure of daily mathematical practice.

Related Topics:
Relations - Functions - Integers - Rational - Real - Complex numbers - Natural numbers

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It is often asserted that axiomatic set theory is thus an adequate foundation for current mathematical practice, in the sense that in principle all proofs produced by the mathematical community could be written formally in set theory terms. It is also generally believed that no serious advantage would come from doing that, in almost all cases: the axiomatic foundations normally used are sufficiently closely aligned to the underlying set theory, that full axiomatic translation yields only a little extra, compared to argument in the usual, traditional informal style. One area where a gap can appear between practice and easy formalisation is in category theory, where for example a concept like 'the category of all categories' requires more careful set-theoretic handling.

Related Topics:
Mathematical practice - Category theory

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