Infinity and vacuous names

Rather than just picking holes in Speranza's posts on infinity I thought I should say something more constructive.

Speranza raised the question how one would add an axiom of infinity into Grice's formal treatments relating to vacuous names, so I had a look back at my rehash of this material to see how that might work.

Before coming to Grice its worth pointing out some different kinds of axiom of infinity.

Speranza discussed infinity in connection with "finitism", and the definition of a finitist (at Wikipedia) is "only accepts the existence of finite mathematical objects".
 This means that the axiom of infinity in NBG which Speranza discussed would not be acceptable to a finitist.

"The" axiom of infinity also arises in the context of Russell and Whitehead's Principia Mathematica.
To do arithematic in Russell's Theory of Types you need an axiom of infinity, and it is a curiosity that Russell remained a logicist even though he considered the axiom of infinity used in Principia to be contingent.
The need of an axiom of infinity (or something equivalent) for doing arithmetic, is not peculiar to Russell's Theory of Types, is it common to all treatments of arithmetic including those approved by finitists (such as PRA), so how can this be if finitists reject the axiom of infinity?

The answer is that there are two quite different things which an axiom of infinity might do.
In the case of ZFC, NBG and most set theories, the axiom of infinity asserts the existence of a set with infinitely many elements.
To do arithemetic we do need infinitely many numbers, but  we don't necessarily have to have them collected together into a set.
In PA and in PRA, and in most first order formalisations of arithmetic, you have an axiom of infinity or its equivalent, but you don't have any infinite objects.
In fact the Peano axioms can be read as a way of asserting that there are infintely many things.

These give us another general formula for asserting that there are infinitely many things which does not require the existence of sets.
We assert the existence of a distinguished element (think zero) and a one-one function (thing successor) such that the distinguished element is not in the range of the function (zero is not a successor).
This ensures that if as we count from the distinguished element using the function to increment, we get a new object every time and count our way through an infinity of objects (numbers).
[sorry, it does here look like we have asserted the existence of a sucessor function, which itself is an infinite object, but though we make use of it we don't actually assert its existence, and it isn't in the range of our quantifiers, so there is a bit of (logically coherent) fudge going on here]

 Going back to Grice's formal systems and considering the question how to introduce an axiom of infinity, there are therefore a number of choices to be made.

The first is whether we just want there to be infinitely many things so that we could do arithmetic, or whether we want there to be things with infinite extensions (so we don't get confounded with finitists, or perhaps because we think that ordinary language is just as expressive as set theory).
To do the former, we could just add in the peano principles, to do the latter the language of sets and the assertion of a set closed under some successor relationship will do.

That was really the easy question.
Grice's system raised more difficult questions because it has different notions of existence and so when we assert the existence of sets we might or might not be doing something similar to asserting that Pegasus flies and hence, Grice insists, that somehow Pegasus exists.

So here is a question about Grice for Speranza.

When it comes to set theory in the context of his treatment of vacuous names, would Grice be thinking of sets as "logical fictions", and hence want to treat sets as non-denoting, or do we want sets to be more solid than that.
One issue which arises in this context is whether one can pile fictions on top of each other in the way that sets can be made from sets.
Can on have a fiction which is build out of fictions in some way?
Is it consistent with Grice's system that not only do sets fail to denote, but that their members lack denotations as well?

I'm afraid I have lost my grip on the system since it is years since I did my rehash (see: http://rbjones.com/rbjpub/pp/doc/t037.pdf) so I would have to spend some time getting it back into my head before I could see what the options are.

Meanwhile Speranza has launched us into extensionalism and other minimalisms (though perhaps he doesn't have more than one) which are an important area in which one might imagine Grice to have serious issue with Carnap, but in which I persist in thinking that Grice's does not see all the possibilities and that not all notions of extensionalism and not all practice of minimalism falls prey to his critiques, or ought by him to be considered objectionable.
So I am more inclined to chase this hare than get deeper into how infinity might work for Grice, which probably does get very complicated.

RBJ