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

[Torkel Franzen <***>]

glen e. p. ropella writes:

>At least in B&M, there is an attempt to treat circularity, in general, 
>referring to mathematical efficacy and modeling as well as logic and 
>philosophy.  In particular, that proposition 2.6 seems to treat several 
>kinds of circularity.  (I emphasize "seems".)

  Non-well-founded set theory is useful in modeling non-well-founded
structures and phenomena.

>In any case, knowing when cycles of these types should and should not be 
>prevented by underlying assumptions is important, regardless of what 
>terms we use to refer to them.  So, we don't have to refer to the whole 
>set as "impredicativities" (even though I still believe that they all 
>satisfy Torkels' definition of "impredicativity as used in logic and 
>philosophy").

  It's unclear to me what it is you believe. Using AFA we prove the
existence of a unique set a such that a={a}. What impredicative
definition do you associate with this set? In the other direction,
do you associate some non-well-founded structure with the impredicative
comprehension principle I stated earlier?

> >It all becomes
> >too much like reflections along the lines of "The Heisenberg
> >uncertainty that pervades social relations means that we cannot
> >simultaneously measure the quantum of intimacy and the momentum
> >of social distance".

>That's hilarious!  Have you actually seen that used? 

  No, I introduced it as an analogy.

  Regarding impredicativity: I have emphasized the basical logical usage,
as illustrated by the impredicative comprehension axiom in second order
arithmetic. But of course there are many philosophical, historical, and
mathematical questions, problems, concepts and results associated with
the concept of predicativity in logic. Anybody with a serious interest
in the subject should at least have a look at the article Predicativity,
by Solomon Feferman, no 32 on this page:

http://math.stanford.edu/~feferman/papers.html