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

[Calvin Ostrum <***>]

On 8/9/05, Torkel Franzen <***> wrote:

> Thus, suppose we define a set A of natural numbers by
>
>   k is in A if and only if k is in every set of natural numbers
>   that contains 5 and is closed under the operation x->x+8.
>
> Since we are here defining A by quantifying over a collection of sets
> containing A, the definition is impredicative. You will notice that
> there is no self-application here, no set of all sets or anything
> of the kind.

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.

> In this case, A can also be defined by a purely arithmetical condition
> (illustrating that impredicativity is not a property of objects, but of
> definitions).

But this does seem to suggest a natural definition of impredicative
object: one all of whose definitions are impredicative.
Although perhaps that definition then has no instances?

> >Perhaps, then we can get back to
> >RR and his claim that "organisms are full of such impredicativities".
>
> That's a different matter. I suspect that Rosen's reference to
> "impredicativities" is best understood as loosely based on some
> perceived analogy with (what he took to be) the logical usage, so that
> studying impredicativity in the sense that the term is actually used in
> logic and philosophy is of no help in understanding his writings.

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?