Implementation of mathematics in set theory: Difference between revisions

Revision as of 21:32, 21 June 2018

Revision as of 23:28, 19 August 2018

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The usual operations of arithmetic can be defined recursively and in a style very similar to that in which the set of natural numbers itself is defined. For example, + (the addition operation on natural numbers) can be defined as the smallest set which contains <math>((x,\emptyset),x)</math> for each natural number <math>x</math> and contains <math>((x,y \cup \{y\}),z \cup \{z\})</math> whenever it contains <math>((x,y),z)</math>.

In NFU, it is not obvious that this approach can be used, since the successor operation <math>y \cup \{y\}</math> is unstratified and so the set ''N'' as defined above cannot be shown to exist in NFU (it is interesting to note that it is consistent for the set of finite von Neumann ordinals to exist in NFU, but this strengthens the theory, as the existence of this set implies the Axiom of Counting (for which see below or the [[New Foundations]] article)).

The standard definition of the natural numbers, which is actually the oldest [[set-theoretic definition of natural numbers]], is as equivalence classes of finite sets under equinumerousness. Essentially the same definition is appropriate to [[New Foundations|NFU]] (this is not the usual definition, but the results are the same): define ''Fin'', the set of finite sets, as