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

[Torkel Franzen <***>]

glen e. p. ropella writes:

>Sorry for my incompetence, here; but, that's not very clear.  Are you 
>talking about things like "the set of all sets"?  Or are you talking 
>about objects that can predicate themselves (e.g. "unity of unity")?

No, that's not it. First, objects are not impredicative, but
definitions or specifications of objects. 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.

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 the general impredicative set comprehension
principle (here: for sets of natural numbers) has no predicative
justification. This is the principle

    There is a set A containing those and only those natural
    numbers k for which F(k),

where F(k) is any condition on k that can be formulated using arithmetical
and logical language, including quantification over sets of natural
numbers.

Kercel's supposed explanation is best set aside. Such remarks as

>Since the collection possesses the same property as the objects in the 
>collection, the collection is an impredicative object.

are irrelevant to the concept of impredicativity as understood in
logic and philosophy, and considered as comments on that concept they are
just confused.

The connection with the paradoxes is just that Russell thought of the
restriction to predicative definitions as a way of blocking the
derivation of paradoxes. But in fact it goes very much further than
that, and explaining what we mean by "set" with reference to the
(extremely impredicative) iterative conception is quite sufficient to
dispose of the set-theoretical paradoxes.

>I agree with you here.  As long as a concept is useful, mathematicians 
>will use it.

Well, mathematicians don't use the concept of impredicativity and
have no idea what it means. What they do is use set existence
principles which are described in logic and philosophy as impredicative.

>But, later in "Vicious Circles" Barwise talks about the iterative 
>conception, if one assumes the foundation axiom, all of the elements of 
>a set must be pre-defined (without referring to sets).

This is no part of the iterative conception of sets. Are you sure
Barwise was not talking rather of the constructive hierarchy?

>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.