# 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 Harvey,

I looked a bit at your RSL paper with interest – linked at your website. A rather startling evolution over your career!

I hope there’s been ‘evolution’ (assuming ‘evolve’ implies ‘improve’)! But it’s surely true that the view has acquired more moving parts. The only actual discontinuity, though, is the one I described between Realism and Naturalism.

One thing that I see in all your writings is a blackboxing of mathematical activity – which can be construed as an uncritical acceptance of the mathematical community to pursue what it feels is important.

This isn’t quite right. As came up in the exchange with Sy, my claim isn’t just that set theorists don’t add V = L to their list of axioms, but that they’re right not to do so — for identifiable mathematical reasons. Similarly, back when the concept of ‘group’ was being isolated, it would have been a bad idea to stick with cancellation laws (instead of inverses) because all the important connections with infinite groups would have been missed.

As to what’s important (or deep), well, do you think there’s a difference between what the community thinks is important and what’s really is important? Do you think the community could overlook something important or think something is important when it’s not? I admit that I do think these things, at least tentatively. (This was one of the questions explored in that workshop on depth.)

In light of your “conversion”, I would like to get your take on whether we are witnessing an emerging “foundational crisis” or whether this is best viewed as ordinary business as usual.

I don’t know that I’d call it a crisis. I think we’ve been forced to face the fact that ‘right’ and ‘wrong’ in mathematics is a more subtle business than we might prefer, given a choice, but physicists have managed to cope with such disappointments in the past and I’m sure we will, too. This might mean that lots of mathematicians prefer not to work in areas like higher set theory, but we can hope its attractions are enough to keep it from dying out.

All best,
Pen

# Re: Paper and slides on indefiniteness of CH

Dear Bob,

What is the precise definition of “maximality”. Is it evident from that definition, that maximality implies there are reals not in HOD? If not, can you give a cite as to where this is proved?

I do not know of a precise definition of “maximality”. Rather I regard “maximality” as an “intrinsic feature” of the universe of sets via what Pen has referred to as “the usual kind of conceptualism: there is a shared concept of the set-theoretic universe (something like the iterative conception); it’s standardly characterized as including ‘maximality’, both in ‘width’ (Sol’s ‘arbitrary subset’) and in the ‘height’ (at least small LCs).” [Pen: I dropped the bit about "reflection" as I wasn't sure what you meant; but I don't think that its omission will affect this discussion.]

“Maximality” is indeed formulated mathematically in a number of different ways in the HP, but I don’t know if there is an ultimate mathematical formulation which fully captures it and therefore cannot claim that there will be a precise definition of this intrinsic feature.

Nevertheless I do regard the existence of reals not in HOD to be derivable from “maximality” for the following reason, which I expect to be shared by others who share the maximal iterative conception: Part of this conception is that the powerset of omega consists of arbitrary subsets of omega. This is violated by V = L, which insists that the only subsets of omega are those which are predicatively definable relative to the ordinals, and also by V = HOD, which insists that the only subsets of omega are those which are definable relative to ordinals.

Returning to V = Ultimate L: I would expect that anyone claiming this to be true would also claim V = L to be true had Jack succeeded in showing that $0^\#$ cannot exist. But V = L is in my view clearly wrong (irregardless of whether $0^\#$ can exist), as well-expressed by Gödel:

“From an axiom in some sense opposite to [V=L], the negation of Cantor’s conjecture could perhaps be derived. I am thinking of an axiom which … would state some maximum property of the system of all sets, whereas [V=L] states a minimum property. Note that only a maximum property would seem to harmonize with the concept of set …”

This argument seems to refute V = Ultimate L.

Best,
Sy