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

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

Calvin Ostrum wrote:
> On 8/10/05, Torkel Franzen <***> wrote:
> > gepr>I am unclear on why B&M would want to swap out the FA for what they call
> > gepr>the "Anti-Foundation Axiom" in order to avoid the situation obtained in
> > gepr>prop 2.6.
> tf>  Because they are writing about the specific subject of non-well-founded
> tf>set theory. It's not a matter of "avoiding a situation", but one of
> tf>pointing out the differences between set theory with foundation and
> tf>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".

Yes.  The larger purpose, as stated by them in the introduction is:

"Indeed, this book [Vicious Circles] is concerned with extending the 
modeling capabilities of set theory to provide a uniform treatment of 
circular phenomena."

The reason I sustained a conversation with Torkel even after he provided 
his personal definition of "impredicativity" was because he seems to 
think that type of circularity is completely different from the other 
types of circularity talked about in Kercel, Kline, B&M, etc.  And I 
don't understand that difference.

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".)

prop 2.6 part 1 "Given FA, for all a, a is not in a" seems to treat the 
most trivial loop.  Part 2, "Given FA, there is no finite sequence a_1, 
a_2, ..., a_n such that a_1 is in a_2 is in ... is in a_n is in a_1", 
seems to treat relational cycles.  Part 3, "Given FA, there are no a, b 
so that a is in TC(b) is in a" seems to treat a kind of circularity 
involving subsets.  (Part 4 is a more complicated form of that type of 
circularity.)

Part 5 is particularly interesting to me:  "Given FA, if c = <a,b>, then 
c != a, c != b, c is not in a, and c is not in b."  I translate this 
into English as:  "a relationship between two sets cannot be identical 
to or contained within either one of those sets".  This is important to 
me because, as a modeler, models can be maps between things (as tight as 
real functions -- M:Input -> Output -- or as loose as simile) and things 
that can be mapped by other models.  In particular, when dealing with 
multi-level or multi-scale models.

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").  And if it causes heartburn for some people, perhaps we 
should just call them all "vicious circles".

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