[FOM] Predicativism and natural numbers

[William Tait]

On Jan 2, 2006, at 2:18 PM, Charles Parsons wrote:
> I think it's a little misleading to talk about
> traditional predicativism as accepting an
> impredicative _definition_ of the natural
> numbers. What one finds in, say, Poincaré and
> Weyl are reasons for _assuming_ the natural
> numbers. So that predicativity in most later
> work, in particular the analyses of Feferman and
> Schuette, is predicativity _given_ the natural
> numers. I think they are quite clear about that.
>
> Several people, myself included, have argued that
> the concept of natural number is impredicative,
> apart from the question of a definition.

In a letter to Goedel in 1970, Bernays speaks of the impredicvativity  
of the concept of number arising from the two roles that numbers  
play: One is that they form a domain N of objects and the second is  
that they act as iterators of operations Phi : D -> D on any domain  
D, including D=N:
n maps to Phi^n : D ->D.

I believe that this is related to Nelson's impredicativity (to which  
Charles refers in his posting), but it also does not really depend  
upon the definition of N as the intersection of hereditary sets.

Bill Tait