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

[Richard G Heck]

Randall Holmes wrote:

>The definition of the natural numbers in ZF does not require the axiom
>of infinity. [snip] However, the definition [of natural number] will not make sense unless the axiom of infinity is true. If the axiom of infinity is not true..., every object is a natural number....
>
Something stronger is true: PA is interpretable in Z(F)-Inf (and, 
famously, conversely). One does not need the axiom of infinity to define 
the natural numbers in a reasonable fashion, meaning: in such a way that 
one can prove their basic properties, viz, those formalized in PA. There 
are several ways of doing this, some of the most famous of which trace 
to work by Wang and Dummett in the 1950s and about which others would be 
more qualified to speak than I.

Richard Heck