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

[Torkel Franzen <***>]

"glen e. p. ropella" <***> says:

>I think both of your descriptions are misleading.  Here's the clearest
>definition I've found:

>"The Model-Based Mind" by Kercel, VanHoozer, and VanHoozer, citing
>Kleene, presents it as:

>"An impredicative property, P(x), of an object x in X, is the property
>such that X is the set of objects possessing property P(x).  In other
>words, an impredicative property participates in its own definition.
>Mathematicians do not deny the existence of impredicativities, but
>regard them as a necessary evil.  Impredicativite processes, closed
>loops of causality and bizarre systems are equivalent concepts."

  I guess it's inevitable that the logical term "impredicative" should
inspire all sorts of extended or analogous uses. The above, however,
is not at all correct applied to the use of "impredicative" in logic.
An impredicative definition, in logic, is one that defines an object -
typically a set - by means of quantification over a totality to which
that object itself belongs. That "mathematicians regard the existence
of impredicativities as a necessary evil" is an arbitrary statement.
Mathematicians in general pay no attention to whether or not a
definition is impredicative, and are unaware of the concept.

>Problems with (including people fixated on) impredicativity are just
>evidence the person is trying to reduce everything to first order logic
>or, as Barwise puts it, the iterative conception, where something can
>only be defined in terms of previously defined things.

  Since you refer to Barwise, I suspect that what you have in mind here
is rather the distinction between well-founded and non-well-founded
sets. This is a different matter. The classical iterative conception
of the set-theoretical universe is profoundly impredicative, i.e.
presupposes that impredicative definitions make good sense.