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

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

Torkel Franzen wrote:
> glen e. p. ropella says:
>
> > Well, it seems to me that a statement like "let a be a set such that 
> > a={a}" is an impredicative definition.
>
>   Why is that?

Because it defines an object by means of a quantification over a 
totality to which the object itself belongs. [grin]

> Do do you associate some non-well-founded structure with
> the impredicative comprehension principle I stated earlier?

Yes.  It seems to me that using the AFA, when defining structures, 
allows impredicative definitions.

> > > This would clarify what sort of poverty of entailment is intended, and
> > > allow readers to judge for themselves the aptness and significance of
> > > "very little", "almost every", and so on.
>
>   The passage you quote makes perfect sense, but has no bearing on
> my question about Rosen's statements.

It does have bearing on your question and Rosen's statements... you just 
don't see it, I guess.  B&M are saying that ZFA is richer than ZFC.  And 
they give a precise statement of that difference.  When RR talks of 
poverty of entailment when using formal systems that don't allow cycles, 
he is _also_ talking about formulating an axiomatic system that allows 
cycles and how that latter system will be richer than the former.

Granted, this has little to do with Goedel's theorem or how it might 
show the poverty of entailment of formal systems.  But, the general gist 
is the same... B&M outline a path from an axiomatic system that is 
limited in its ability to deal with cycles to one that is less limited. 
 RR was attempting to do a similar thing.  Denying that they are 
similar efforts is simply belligerent.

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