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

[Torkel Franzen <***>]

Calvin Ostrum writes:

>However, we will also notice that the set A already existed before
>this definition was given.  So in some sense, the definition does
>nothing but pick out the set.   I think the sort of thing people
>are concerned about here will involve a different sort of
>"definition" akin more to a "construction" in which the
>definition in some sense has the power to "bring into being",
>rather than just "pick out", the pre-existing object in question.

  Such ideas are indeed relevant to the notion of impredicativity
in logic.

>But this does seem to suggest a natural definition of impredicative
>object: one all of whose definitions are impredicative.

  This involves the highly problematic notion of "all definitions".
Through somewhat involved logical definitions and investigations,
it leads to the ordinal Gamma_0 as the limit of predicativity in
the hierarchy of constructible sets.

>Is there any possibility that Rosen's use of "impredicativity" can be
>made at all precise?  Surely there is more to it than loose analogy?

  As to that, I cannot say.