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

[Calvin Ostrum <***>]

On 8/10/05, Torkel Franzen <***> wrote:

> >I am unclear on why B&M would want to swap out the FA for what they call
> >the "Anti-Foundation Axiom" in order to avoid the situation obtained in
> >prop 2.6.
> 
>   Because they are writing about the specific subject of non-well-founded
> set theory. It's not a matter of "avoiding a situation", but one of
> pointing out the differences between set theory with foundation and
> with anti-foundation.

But surely that is not the only reason.   Why would they bother to
write about such a thing, after all?  I believe it is largely because
they believe the non-well-founded set theories give particularly
good treatments of such things as the Liar's Paradox and other
kinds of troublesome forms of self-reference (at least that is
probably true of Barwise's earlier book "The Liar: An Essay on
Truth and Circularity".

Anyway, in the case of Rosen, we are not talking merely about
specifying or picking out life in a "circular" manner.   We are
talking about the fact that life "creates itself", so that the
circularity really is part of the construction, not just the 
specification.    But about why this means that life cannot be
computable, or mechanical, or "complex in the Rosean
sense", or whatever, I have no real clue.