[FOM] The definition of the natural numbers vs. the axiom of infinity

[William Tait]

On Thursday, January 2, 2003, at 01:17  PM, Randall Holmes wrote:

> The definition of the natural numbers in ZF does not require the axiom
> of infinity.  A natural number is _defined_ as an object which belongs
> to any set which contains the empty set and is closed under von
> Neumann successor.  One does not need the axiom of infinity for this
> to succeed as a definition.  One does not even need the axiom of
> infinity in order to quantify over all "natural numbers" so defined,
> because in ZF quantifiers do not need to be bounded by sets.
>
> However, the definition will not make sense unless the axiom of
> infinity is true.

Then we should try another one: call a (von Neumann) ordinal of the 
first kind if it has a predecessor or  is 0. A natural number = finite  
ordinal is an ordinal  of the first kind all of whose elements are 
ordinals of the first kind.

Best wishes for the new year,

Bill Tait