[FOM] Predicativism and natural numbers

[Giovanni Lagnese]

Charles Parsons wrote:

> I'm not sure why you say the
> predicativist conception
> of _set_ is impredicative

I mean the definition of set as the smallest collection inductively 
generated by some rules, as in Martin Loef's Intuitionistic (and 
"predicative") Type Theory.
It's necessary to say "the smallest", so the concept of set is 
impredicative.

But the point is...
If there is a philosophical reason according to which impredicative 
definitions/concepts must be rejected, why predicativists can accept *some* 
impredicative definitions/concepts?
In the case of impredicative definitions/concepts they accept, is not valid 
the same philosophical argument by which impredicative definitions/concepts 
must be rejected?

GL