Tag Archives: Leading question in set theory

Re: Paper and slides on indefiniteness of CH

Dear Pen and Hugh,

Pen:

Well I said that we covered everything, but I guess I was wrong! A new question for you popped into my head. You said:

The HP works quite differently. There the picture leads the way — the only legitimate evidence is Type 3. As we’ve determined over the months, in this case the picture involved
has to be shared, so that it won’t degenerate into ‘Sy’s truth’.

I just realised that I may have misunderstood this.

When it comes to Type 1 evidence (from the practice of set theory as mathematics) we don’t require that opinions about what is “good set theory” be shared (and “the picture” is indeed determined by “good set theory”). As Peter put it:

Different people have different views of what “good set theory” amounts to. There’s little intersubjective agreement. In my view, such a vague notion has no place in a foundational enterprise.

I disagree with the last sentence of this quote (I expect that you do too), but the fact remains that if we don’t require a consensus about “good set theory” then truth does break into (“degenerate into” is inappropriate) “Hugh’s truth”, “Saharon’s truth”, “Stevo’s truth”, “Ronald’s truth” and so on. (Note: I don’t mean to imply that Saharon or Stevo really have opinions about truth, here I only refer to what one reads off from their forms of “good set theory”.) I don’t think that’s bad and see no need for one form of “truth” that “swamps all the others”.

Now when it comes to the HP you insist that there is just one “shared picture”. What do you mean now by “picture”? Is it just the vague idea of a single V which is maximal in terms of its lengthenings and “thickenings”? If so, then I agree that this is the starting point of the HP and should be shared, independently of how the HP develops.

In my mail to you of 31.October I may have misinterpreted you by assuming that by “picture” you meant something sensitive to new developments in the programme. For example, when I moved from a short fat “picture” based on the IMH to a taller one based on the \textsf{IMH}^\#, I thought you were regarding that as a change in “picture”. Let me now assume that I made a mistake, i.e., that the “shared picture” to which you refer is just the vague idea of a single V which is maximal in terms of its lengthenings and “thickenings”.

Now I ask you this: Are you going further and insisting that there must be a consensus about what mathematical consequences this “shared picture” has? That will of course be necessary if the HP is to claim “derivable consequences” of the maximality of V in height and width, and that is indeed my aim with the HP. But what if my aim were more modest, simply to generate “evidence” for axioms based on maximality just as TR generates “evidence” for axioms based on “good set theory”; would you then agree that there is no need for a consensus, just as there is in fact no consensus regarding evidence based on “good set theory”?

In this way one could develop a good analogy between Thin Realism and a gentler form of the HP. In TR one investigates different forms of “good set theory” and as a consequence generates evidence for what is true in the resulting “pictures of V”. In the gentler form of the HP one investigates different forms of “maximality in height and width” to generate evidence for what is true in a “shared picture of V”. In neither case is there the presumption of a consensus concerning the evidence generated (in the original HP there is). This gentler HP would still be valuable, just as generating different forms of evidence in TR is valuable. What it generates will not be “intrinsic to the concept of set” as in the original ambitious form of the HP, but only “intrinsically-based evidence”, a form of evidence generated through an examination of the maximality of V in height and width, rather than by “good set theory”.

Hugh:

1. Your formulation of \textsf{IMH}^\# is almost correct:

M witnesses \textsf{IMH}^\# if

1) M is weakly #-generated.

2) If \phi holds in an outer model of M which is weakly
#-generated then \phi holds in an inner model of M.

But as we have to work with theories, 2) has to be: If for each countable \alpha, \phi holds in an outer model of M which is generated by an alpha-iterable presharp then phi holds in an inner model of M.

2. Could you explain a bit more why V = Ultimate L is attractive? You said: “For me, the “validation” of V = Ultimate L will have to come from the insights V = Ultimate L gives for the hierarchy of large cardinals beyond supercompact.” But why would those insights disappear if V is, for example, some rich generic extension of Ultimate L? If Jack had proved that 0^\# does not exist I would not favour V = L but rather V = some rich outer model of L.

3. I told Pen that finding a GCH inner model over which V is generic is a leading open question in set theory. But you gave an argument suggesting that this has to be strengthened. Recently I gave a talk about HOD where I discussed the following four properties of an inner model M:

Genericity: V is a generic extension of M.

Weak Covering: For a proper class of cardinals alpha, alpha^+ = alpha^+ of M.

Rigidity: There is no nontrivial elementary embedding from M to M.

Large Cardinal Witnessing: Any large cardinal property witnessed in V is witnessed in M.

(When 0^\# does not exist, all of these hold for M = L except for Genericity: V need not be class-generic over L. As you know, there has been a lot of work on the case M = \text{HOD}.)

Now I’d like to offer Pen a new “leading open question”. (Of course I could offer the PCF Conjecture, but I would prefer to offer something closer to the discussion we have been having.) It would be great if you and I could agree on one. How about this: Is there an inner model M satisfying GCH together with the above four properties?

Thanks,
Sy

Re: Paper and slides on indefiniteness of CH

Dear Sy,

You wrote to Pen:

But to turn to your second comment above: We already know why CH doesn’t have a determinate truth value, it is because there are and always will be axioms which generate good set theory which imply CH and others which imply not-CH. Isn’t this clear when one looks at what’s been going on in set theory? (Confession: I have to credit this e-mail discussion for helping me reach that conclusion; recall that I started by telling Sol that the HP might give a definitive refutation of CH! You told me that it’s OK to change my mind as long as I admit it, and I admit it now!)

ZF + AD will always generate “good set theory”…   Probably also V = L…

This seems like a rather dubious basis for the indeterminateness of a problem.

I guess we have something else to put on our list of items we simply have to agree we disagree  about.

So the best one can do with a problem like CH is to say: “Based on a certain Type of evidence, the truth value of CH is such and such.” As said above, Type 1 evidence (the development of set theory as an area of mathematics) will never yield a fixed truth value, we don’t know yet about Type 2 evidence (ST as a foundation) and I still conjecture that Type 3 evidence (based on the Maximality of the universe of sets in height and width) will imply that CH is false.

There will never be such a resolution of CH (for the reasons I gave above). The best one can do is to give a widely persuasive argument that CH (or not-CH) is needed for the foundations of mathematics or that CH (or not-CH) follows from the Maximality of the set-concept. But I would not expect either achievement to draw great acclaim, as nearly all set-theorists care only about the mathematical development of set theory and CH is not a mathematical problem.

This whole discussion about CH is of interest only to philosophers and a handful of philosophically-minded mathematicians. To find the leading open questions in set theory, one has to instead stay closer to what set-theorists are doing. For example: Provably in ZFC, is V generic over an inner model which satisfies GCH?

Why is this last question a leading question?  If there is an inner model with a measurable Woodin cardinal it is true, V is a (class) generic extension of an inner model of GCH.

You must mean something else. Focusing on eliminating the assumption of there is an inner model of a measurable Woodin cardinal seems like a rather technical problem.

Regards,
Hugh