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

["glen e. p. ropella" <***>]

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

[grin]  Yes, obviously.

> > 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",

They did not state that FA is "for all a, a is not a member of a".  They 
stated:

   Given FA, for all a, a is not a member of a.

I.e. they stated that the FA implies "for all a, a is not a member of 
a".  I'm not sure where you got the idea that they thought this was a 
statement of the FA.

> 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.

So, then why would you claim that Barwise and Moss, in their statement 
of "proposition 2.6", are grossly incorrect?

--
glen e. p. ropella              =><=                Hail Eris!
H: 503-630-4505                       http://ropella.net/~gepr
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