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

Torkel Franzen wrote: 
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.

OK. I have to paraphrase this to see if I understand. An impredicative definition (of an object, set, predicate, whatever) is one that relies on multiple collections of objects, including the object being defined, where all those collections relate in some way. I.e. collection x has an impredicative definition if that definition depends on some R such that aRb, xRa, and xRb, for collections x, a, and b.

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.

So, if F(k) involves relations between sets, then the definition of A is impredicative. Can you given an example of "predicative justification"? Perhaps that's the missing link.

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

Well, it's hard to set aside a citation that is often used. It's better to clarify what Kercel's motivation was and how it relates to the prior use of the term (even if we come to the opinion that he was misguided). If we just ignore the fact that he uses the term and don't address how it relates to the previous uses, we just stay confused.

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.

OK. [grin] I don't mind admitting that I'm confused. But, the confusion will persist until the language is corrected. Along those lines, since we're discussing circularity, your admonishment that objects are not impredicative but definitions _are_ is only partly helpful. Definitions _are_ objects to some extent. So, what you've told me so far is that only _some_ objects are impredicative. In particular, objects that define other objects are impredicative if they refer to relations between collections, where the object being defined is one of the related collections.

Again, this doesn't seem very different from what Kercel states. The only difference is that Kercel is using P(x) and you're using F(k). And it also doesn't seem that different from what Barwise is claiming in his Proposition 2.6 from Vicious Circles: Given the foundation axiom, for all a, a is not a member of a. (apparently prohibiting impredicative definitions)

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.

Sorry, I'm conflating the points of these two sentences. Are you saying that "restriction to predicative definitions" explains what we mean by "set"?

Or are you saying that "restriction to predicative definitions" (via the definition of "set" and the iterative conception) is sufficient to dispose of the 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.

They may not refer to it using the term "impredicative"; but, I would bet that they use the concept. Of course, arguing about what "mathematicians" do and don't use is pretty silly, anyway. As I said, they'll use any concept that ends up being useful.

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?

All I can do is quote him (and Moss). I have no idea what he was thinking at the time. [grin] His discussion of the iterative conception of a set is in the context of his discussion about the size of classes. He says:

"... by reference to the so-called iterative (or cumulative) conception of set, which says that the only sets there are are those that can be 'collected together' out of 'previously constructed' sets."

Later in that same discussion, he says:

"... now that we are used to the idea that non-wellfounded sets are perfectly sensible objects, we have lost the cumulative picture and so the explanation of the parasoxes is made available."

I'm _pretty_ sure he's talking about "the iterative conception of sets", since he uses that phrase explicitly. And I'm pretty sure this has _something_ to do with definitions. And I'm pretty sure this is related to how things are defined, whether by relations amongst sets, including the thing being defined, or not. Whether there is some subtle distinction between "iterative conception of sets" and "constructive hierarchy" is beyond me.

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.

OK. That helps me to know that RR might have had a _private_ definition of "impredicativity". Does anyone here have an RR-specific definition I can substitute when he uses the word? Judith, yours was a bit too vague... Perhaps you can make it more precise?

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