Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Set theory ZFC foundations for mathematics

Axiomatic set theory - 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.

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:

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

Ordered lists of greater length can be constructed inductively:

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

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.

We proceed inductively:

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

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.

The original set theoretical definition of the natural numbers defined each natural number n as the set of all sets with n elements (this can be managed without the apparent circularity of this brief summary). This definition (due to Frege and Russell) does not work in ZFC because the collections involved are too large to be sets. However, this approach does work in New Foundations and subsystems of NF known to be consistent.

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.

Adapted from the Wikipedia article "Set theory ZFC foundations for mathematics", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki