Dear Pen,

On Sun, 19 Oct 2014, Penelope Maddy wrote:

Dear Sy,

For present purposes, what matters is that set theory has, as one of its goals, the kind of thing Zermelo identifies. This is part of the goal of providing the sort of foundation that Claudio and I were talking about (a kind of certification and a shared arena).

I interpreted the Zermelo quote to mean that ST’s task is to provide a useful foundation for mathematics through a mathematical clarification of ‘number’, ‘order’ and ‘function’, Is that correct? This goal is then Type 2, i.e. concerned with ST’s role as a foundation for mathematics.

Yes, in your classification (if I’m remembering it correctly), this would be a Type 2 goal, that is, a goal having to do with the relations of set theory to the rest of mathematics. (My recollection is that a Type 1 goal is a goal within set theory itself, as a branch of mathematics, and Type 3 is the goal of spelling out the concept of set, regardless of its relations to mathematics of either sort, as a matter of pure philosophy.)

Well, I never talked about “goals” really, but simply about evidence for the truth of set-theoretic assertions. Yes, Type 1 refers ST as a branch of math and Type 2 refers to ST as a foundation for math, with their corresponding forms of evidence. Type 3 concerns evidence for set-theoretic assertions resulting from an analysis of the set-concept, so it is not just a matter of pure philosophy, it is the generation of mathematical assertions which are evidenced through such an analysis.

I don’t see that it’s being Type 2 in any way disqualifies it as a goal of set theory, with attendant methodological consequences.

I agree! I was just trying to understand your use of the word “foundation” in your message.

It’s true that set theory has been so successful in this role and is now so entrenched that it’s become nearly invisible, and neither set theorists nor mathematicians generally give it much thought anymore, but it was explicit early on and it remains in force today (as that recent quotation from Voevodsky indicates).

Good point.

I think it’s fair to say that contemporary set theory also has the goal of resolving CH somehow.

No, Type 1 considerations (ST as a branch of math) are not concerned with resolving CH, that is just something that a handful of set-theorists talk about. The rest are busy developing set theory, independent of philosophical concerns. Both Hugh and I do lots of ST for the sake of the development of ST, without thinking about this philosophical stuff. Philosophers naturally only see a small fraction of what is going on in ST, for the simple reason that 90% of what’s going on does not appear to have much philosophical significance (e.g. forcing axioms).

It seems to me disingenuous to suggest that resolving CH, and devising a full account of sets of reals more generally, is not one of the goals of set theory — indeed a contemporary goal with strong roots in the history of the subject.

Good luck selling that to the ST public. This is interesting to you, me and many others in this thread, but very few set-theorists think it’s worth spending much time on it, let’s not deceive ourselves. They are focused on “real set theory”, mathematical developments, and don’t take these philosophical discussions very seriously.

To say this is in no sense to deny that you and Hugh and other set theorists have many other goals besides. (Incidentally, I don’t see why you think forcing axioms are of no interest to philosophers, but let that pass.)

There are others.

Such as? I think that just as the judgments about “good” or “deep” ST must be left to the set-theorists, perhaps with a little help from the philosophers, so must judgments about “the goals of set theory”.

I haven’t attempted to list other goals because, as a philosopher, I’m not well-placed to do so (as you point out).

The goals I’m aware of that ST-ists seem to really care about are much more mathematical and specific, such as a thorough understanding of what can be done with the forcing method.

Going back to:

“I realize this is frustrating for you, but what you’ve said so far about the ‘reduction to ctms’ hasn’t yet produced understanding (for me) or conviction (for Hugh or Harvey).

I’m still stuck, as above, on figuring out where the ctms live, what makes everything ‘ideal’, and so on. They have other concerns.”

Pen: If you have questions about this, please pose them.

Harvey: I repeat the argument below (for the 5th time? Harvey, please read these e-mails!). As a mathematician you should have no trouble following it.

It is what I wrote to Pen on 13.October:

Maybe this will help: If a first-order sentence holds in all countable transitive models of ZFC then it holds in all transitive models of ZFC. That is LS (Löwenheim-Skolem). Now we want the same for “maximal” models (if a first-order sentence holds in all “maximal” countable transitive models of ZFC then it holds in all “maximal” transitive models of ZFC). Now this is not obvious because “maximality” in its various forms is not first-order. And this kind of transfer from countable to arbitrary doesn’t work for second-order sentences. But the point of my mail to Geoffrey is that “maximality” as treated in the HP is only slightly worse than first-order, it is first-order in a “slight lengthening” of the model, and this is good enough to apply LS again.

More details: Take the IMH as an example. It is expressible in V-logic. And V-logic is first-order over the least admissible (Goedel-) lengthening of V (i.e. we go far enough in the L-hierarchy built over V until we get a model of KP). We apply LS to this admissible lengthening, that’s all.

It is not a hard argument, so I don’t know what all the fuss is here. Of course I’m willing to answer any questions but I don’t want to just keep repeating myself!

Hugh hasn’t complained about the reduction to the Hyperuniverse, but for some reason he feels that this reduction is of no importance and has decided that all that matters is the end-product of that reduction, an analysis of ctm’s. Then he goes further and makes no distinction between the study of maximality criteria for ctm’s and the much bigger study of ctm’s in general! I really don’t understand why Hugh makes these moves. As with Harvey, anything I can say about this would just be a repetition of what I have already said many times.

Pen, can you be of some help here? How are your “negotiating skills”?

Thanks,

Sy