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== Finite sets and natural numbers ==
Natural numbers can be considered either as finite ordinals or finite cardinals. Here
we consider them as finite cardinal numbers. This is the first place where a major
difference between the implementations in [[ZFC]] and [[NFU]] becomes evident.
The Axiom of Infinity of [[ZFC]] tells us that there is a set A which contains <math>\emptyset</math>
and contains <math>y \cup \{y\}</math> for each <math>y \in A</math>. This set A is not uniquely
determined (it can be made larger while preserving this closure property): the set N of natural
numbers is
*<math>\{x \in A \mid (\forall B.(\emptyset \in B \wedge (\forall y.y \in B \rightarrow y \cup \{y\} \in B) \rightarrow x \in B)\}</math>
which is the intersection of all sets which contain the empty set and are closed under
the "successor" operation <math>y \mapsto y \cup \{y\}</math>.
In [[ZFC]], we say that a set <math>A</math> is finite iff there is <math>n \in N</math> such
that <math>|n|=|A|</math>: further, we define <math>|A|</math> as this n for finite A. (It can
be proved that no two distinct natural numbers are the same size).
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>((\emptyset,\emptyset),\emptyset)</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 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.
We here present the definition of N appropriate to [[NFU]] in exactly this way (this is not
the usual way to do it, but the results are the same): define Fin, the set of finite sets,
as
*<math>\{A \mid (\forall F.(\emptyset \in F \wedge (\forall xy.x \in F \rightarrow x \cup \{y\} \in F)) \rightarrow A \in F)\}</math>
For any set <math>A \in Fin</math>, define <math>|A|</math> as <math>\{B \mid A \sim B\}</math>.
Define N as the set <math>\{|A| \mid A \in Fin\}</math>.
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