Tag Archives: Descriptive set theory

Re: Paper and slides on indefiniteness of CH

We haven’t discussed Hugh’s Ultimate L program much. There are two big differences between this program and CTMP (aka HP). As I understand it,

1. It offers a proposed preferred set theoretic universe in which it is clear that CH holds – but the question of its existence relative to large cardinals (or relative consistency) is a (or the) major open question in the program.

2. In connection with 1, there are open conjectures (formulated in ZFC) which show how to refute Reinhardt’s axiom (the existence of j:V \to V) within ZF (and more).

So even if one rejects 1, this effort will leave us at least with 2, which is nearly universally regarded as important in the set theory community.

It would be nice for most people on this thread to have a generally understandable account of at least the structure of 1. I know that there has been some formulations already on the thread, but they are a while ago, and relatively technical. So let me ask some leading questions.

Can this Ultimate L proposal be presented in the following generally understandable shape?

Goedel’s constructible sets, going by the name of L, are built up along the ordinals in a very well defined way. This allows all of the usual set theoretic problems like CH to become nice mathematical problems, when formulated within the set theoretic universe of constructible sets, L. Thanks to Goedel, Jensen, and others, all of these problems have been settled as L problems. (L is the original so called inner model).

Dana Scott showed that L cannot accommodate measurable cardinals. There is an incompatibility.

Jack Silver showed that L can be extended to accommodate measurable cardinals. He worked out L[U], where U stands for a suitable measure on a measurable cardinal. The construction is somewhat analogous to the original Goedel’s L. Also all of the usual set theoretic problems like CH are settled in L[U].

This Gödel-Silver program (you don’t usually see that name though) has been lifted to considerably stronger large cardinals, with the same outcome. The name you usually see is “the inner model program”. The program slowed down to a trickle, and is stalled at some medium large cardinals considerably stronger than measurable cardinals, but very much weaker than – well it’s a bit technical and I’ll let others fill in the blanks here.

“Inner model theory for a large cardinal” became a reasonably understood notion at an informal or semiformal level. And some good test questions emerged that seem to be solvable only by finding an appropriate “inner model theory” for some large cardinals.

So I think this sets the stage for a generally understandable or almost generally understandable discussion of what Hugh is aiming to do.

Perhaps Hugh has picked out some important essential features of what properties the inner models so far have had, adds to them some additional desirable features, and either conjectures or proves that there is a largest such inner model – if there is any such inner model at all. I am hoping that this is screwed up only a limited amount, and the accurate story can be given in roughly these terms, black boxing the important details.

There are also a lot of important issues that we have only touched on in this thread that I think we should return to. Here is a partial list.

1. Sol maintains that there is a crucial difference between (\mathbb N,+,\times) and (P(\mathbb N),\mathbb N,\in,+,\times) that drives an enormous difference in the status of first order sentences. Whereas Peter for sure, and probably Pen, Hugh, Geoffrey strongly deny this. I think that Sol’s position is stronger on this, but I am interested in playing both sides of the fence on this. In particular, one enormous difference between the two structures that is mathematically striking is that the first is finitely generated (even 1-generated), whereas the second is not even countably generated. Of course, one can argue both for and against that this indisputable fact does or does not inform us about the status of first order sentences. Peter has written on the thread that he has refuted Sol’s arguments in this connection, and Sol denies that Peter has refuted Sol’s arguments in this connection. Needs to be carefully and interactively discussed, even though there has been published stuff on this.

2. The idea of “good set theory” has been crucial to the entire thread here. Obviously, there is the question of what is good set theory. But even more basic is this: I don’t actually hear or see much done at all in higher set theory other than studies of models of higher set theory. By higher set theory I mean more or less set theory except for DST = descriptive set theory. See, DST operates just like any normal mathematical area. DST does not study models of DST, or models of any set theory. DST basically works with Borel and sometimes analytic sets and functions, and applies these notions to shed light on a variety of situations in more or less core mathematics. E.g., ergodic theory, group actions, and the like. Higher set theory operates quite differently. It’s almost entirely wrapped up in metamathematical considerations. Now maybe there is a point of view that says I am wrong and if you look at it right, higher set theorists are simply pursuing a normal mathematical agenda – the study of sets. I don’t see this, unless the normal mathematical area is supposed to be “the study of models of higher set theory”. Perhaps people might want to interpret working out what can be proved from forcing axioms? Well, I’m not sure this is similar to the situation in a normal area of mathematics like DST. So my point is: judging new axioms for set theory on the basis of “good set theory” or “bad set theory” doesn’t quite match the situation on the ground, as I see it.

3. In fact, the whole enterprise of higher set theory has so many features that are so radically different from the rest of mathematics, that the whole enterprise, to my mind, should come into serious question. Now I want to warn you that I am both a) incredibly enthusiastic about the future of higher set theory, and b) incredibly dismissive about any future of higher set theory whatsoever — all at the same time. This is because a) is based on certain special aspects of higher set theory, whereas b) is based on the remaining aspects of higher set theory. So when you see me talking from both sides of my mouth, you won’t be shocked.