Re: defn of impredicativity (was Re: goals and language?)

[Torkel Franzen <***>]

glen e. p. ropella writes:

>Unfortunately, the only reason I 
>care at all about the term "impredicativity" is because RR used it in 
>some places.

  My suggestion then is that you try to find out what *he* meant by it,
by studying his writings. 

>> Such comments as
>> 
>>>Given the foundation axiom, for 
>>>all a, a is not a member of a.  (apparently prohibiting impredicative 
>>>definitions)
>> 
>> are grossly incorrect.

>Well, then perhaps Barwise and Moss are grossly incorrect.

  There is no reason to think so. Barwise and Moss were perfectly aware
that

1) The foundation axiom is not "for all a, a is not a member of a",
but the stronger, purely set-theoretical statement "for every non-empty set b,
there is an x in b such that the intersection of b and x is empty",

and

2) the foundation axiom is perfectly compatible with impredicative
comprehension axioms. In particular, the impredicative axiom for the
existence of sets of natural numbers that I stated is fully compatible
with the foundation axiom, as is clear from the fact that it involves
only numbers and sets of numbers.