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

Torkel Franzen wrote:
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.


Ahhh, I get it. OK. You're simply schooling me on what words to use to successfully communicate with you (or some larger community isolated to "logic and philosophy"). I didn't get that before. Cool.

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?

Well, it seems to me that a statement like "let a be a set such that a={a}" is an impredicative definition. Perhaps I'm wrong. But, if I'm not, it seems to me that I can say a is "an impredicatively defined object". Then as a linguistic shortcut (which you cannot avoid if you use English to talk about things), it seems reasonable to say things like "a is an impredicative set". This sort of contraction and neologism happens in natural languages all the time. And it seems unreasonable to hijack every conversation or criticize every author who takes such shortcuts. Of course, I agree that when they switch from English to math (or logic), then they have to abandon such shortcuts.

In the other direction,
do you associate some non-well-founded structure with the impredicative
comprehension principle I stated earlier?

Yes. It strikes me that using the AFA allows impredicative definitions.

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

No, I introduced it as an analogy.

Damn. I was hoping you had some example like those where autopoiesis or autocatalysis is used in law and social systems.

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

Thanks for that link.

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glen e. p. ropella              =><=                Hail Eris!
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