# Re: Paper and slides on indefiniteness of CH

Dear Pen,

On Tue, 19 Aug 2014, Penelope Maddy wrote:

Dear Sy,

I think anyone would be nervous about linking set-theoretic truth to a concept private to one person and (perhaps) a handful of his co-workers,

I am disappointed to hear you say this.

I apologize for disappointing you. I was going by some of what you’ve written in these exchanges. First this:

First question: Is this your personal picture or one you share with others?

I don’t know, but maybe I have persuaded some subset of Carolin Antos, Tatiana Arrigoni, Radek Honzik and Claudio Ternullo (HP collaborators) to have the same picture; we could ask them.

Why do you ask? Unless someone can refute my picture then I’m willing to be the only “weirdo” who has it.

This was very badly said, my mistake. Of course I expect that other people’s mental pictures of the universe of sets share a lot in common with mine. When I made these remarks I failed to understand the importance of your questions for grounding my notion of “intrinsic features of V”. Sorry, philosophically I still have much to learn.

You then softened this with:

You got this wrong. I indeed expect that others have similar pictures in their heads but can’t assume that they have the same picture. There is Sy’s picture but also Carolin’s picture, Tatiana’s picture … Set-theoretic truth is indeed about what is common to these pictures after an exchange of ideas.

I assumed you still meant to limit the range of people whose pictures are relevant to a fairly small group.

No, the “…” also includes Pen’s picture!

Otherwise the collection of things ‘common to these pictures’ would get too sparse.

In any case, these further explanations are most helpful …

2. Mental pictures

Each set-theorist who accepts the axioms of ZFC has at any given time an individual mental picture of the universe of sets.

OK. Everyone has his own concept of the set-theoretic universe.

3. Intrinsic features of the universe of sets

These are those practice-independent features common to the different individual mental pictures, such as the maximality of the universe of sets. Thus intrinsic features are determined by the set theory community. (Here I might lose people who don’t like maximality, but that still leaves more than a handful.)

OK. Intrinsic features are those common to all the concepts of set-theoretic universe (or close enough).

With the important requirement of “practice-independence”! PD and large large cardinal existence don’t sneak in (unless they are at some point derivable from intrinsically-based criteria).

So far, this seems to be 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). Also reflection (see below).

“Reflection” is ambiguous; I use it below to pass from “features” to “criteria” but maybe a better word there would be “mirroring” or something like that. Usually I use “reflection” to refer to reflection principles, i.e. ordinal-maximality.

4. The Hyperuniverse

This mathematical construct consists of all countable transitive models of ZFC. These provide mathematical proxies for all possible mental pictures of the universe of sets. Not all elements of the Hyperuniverse will serve as useful proxies as for example they may fail to exhibit intrinsic features such as maximality.

OK. We stipulate that the hyperuniverse contains all CTMs of ZFC. But some of these (only some? — they’re all countable, after all) fail to exhibit maximality, etc.

The “maximality” of our mental pictures of the universe is mirrored by the countable models which satisfy mathematical criteria which are faithfully derived from the feature of
“maximality”. For example, the minimal model of set theory will satisfy none of the criteria based on “maximality”, but a countable model of V = L could satisfy an ordinal-maximality criterion (it could be “tall” in terms of reflection but still countable). There also could be countable models that satisfy the IMH, a “powerset” maximality criterion. You are right, you lose “literal maximality” when you look at countable models but you can still faithfully mirror “maximality” using countable models. Remember we are in the end after first-order consequences which don’t notice if the model is countable or not. And the huge new advantage of working with countable models as “proxies” is that we have the ability to generate and compare universes, allowing us to express “external forms” of “maximality” in ways that were not derivable from the old maximal iterative conception. This is not possible using uncountable models.

5. Mathematical criteria

These are mathematical conditions imposed on elements of the Hyperuniverse which are intended to reflect intrinsic features of the universe of sets. They are to be unbiased, i.e. formulated without appeal to set-theoretic practice. A criterion is intrinsically-based if it is judged by the set theory community to faithfully reflect an intrinsic feature of the universe of sets. (There are such criteria, like reflection, which are judged to be intrinsically-based by more than handful.)

OK. Now we’re to impose on the elements of the hyperuniverse the conditions implicit in the shared concept of the set-theoretic universe. These include maximality, reflection, etc. (We’re weeding the hyperuniverse, right?)

Right!

6. Analysis and synthesis

An intrinsic feature such as maximality can be reflected by many different intrinsically-based mathematical criteria. It is then important to analyse these different criteria for consistency and the possibility of synthesizing them into a common criteria while preserving their original intentions. (I am sure that more than a handful can agree on a suitable synthesis.)

I think this is the key step (or maybe it was (5)), the step where the HP is intended to go beyond the usual efforts to squeeze intrinsic principles out of the familiar concept of the set-theoretic universe. The key move in this ‘going beyond’ is to focus on the hyperuniverse as a way of formulating new versions of the old intrinsic principles.

Let me stop at this point, because I’m afraid my paraphrase has gone astray. You once rejected the bit of my attempted summary of your view that said the new hyperuniverse principles ‘build on’ principles from the old concept of the set-theoretic universe, and I seem to have fallen back into that misunderstanding. The old concept you characterize as ‘just the maximal iterative conception’. (You don’t include maximizing ‘width’ in this, though I think it is usually included.)

I just wasn’t sure whether the phrase “maximal iterative conception” includes maximising width; if so, fine.

I’m not sure how to describe the new concept, but the new principles implicit in it are different in that ‘they deal with external features of universes and are logical in nature’ (both quotes are from your message of 8/8).

What I’m groping for here is a characterization of where the new intrinsic principles are based. It has to be something other than the old concept of the set-theoretic universe, the maximal iterative conception. I keep falling into the idea that the new principles are generated by thinking about the old principles from the point of view of the hyperuniverse, that the new principles are new versions of the old ones and they go beyond the old ones by exploiting ‘the external features of universes’ (revealed by the hyperuniverse perspective) in logical terms. But this doesn’t seem to be what you want to say. Is there a different, new concept, with new intrinsic principles?

I think we are in agreement, the problem was my failure to realise that the “maximal iterative conception” does indeed include maximising width. So keeping my terminology, it’s the same old feature of “maximality” but the mathematical criteria derived from this “go beyond the old [internal] ones by exploiting the external features of universes revealed by the
hyperuniverse perspective in logical terms”, just as you have said.

An aside: as I understand things, it was the purported new concept that seemed to threaten to be limited to a select group. If the relevant concept in all this is just the familiar concept of the set-theoretic universe — which does seem to be broadly shared, which conceptualists generally are ready to embrace — and the hyperuniverse is just a new way of extracting information from that familiar concept, then at least one of my worries disappears.

That is exactly right. One less thing to worry about!

Thanks a lot,
Sy

PS: Obviously my immediate goal is to get to the point where I can convince you that the programme is on a solid foundation, even if you are not especially interested in it (which is OK, as I am not tying this to practice and I know how strongly you feel about practice). But at least with some reassurance of a solid philosophical foundation I will feel a lot better about devoting myself to the hard mathematics necessary to implement the programme. So please keep challenging me and looking for “cracks” in the foundation!

# Re: Paper and slides on indefiniteness of CH

Dear Hugh,

On Sun, 17 Aug 2014, W Hugh Woodin wrote:

Well, it is hard not to respond. So I guess I will violate my “last message” prediction. Hopefully my other predictions are not so easily refuted. My apologies to the list. Never say never I suppose.

Hugh:

Regarding your two points directed at me and the HP:

1) Having established Con LC, one has established that every set X belongs to an inner model in which LC is witnessed above X.

I don’t necessarily agree. There are only countably many LC axioms and a perfectly coherent scenario is that each holds in an inner model which is coded by some real. That demands only countably many reals and in no way suggests that there should be large LCs in inner models containing all of the reals.

But please don’t misunderstand me: The HP is a programme for discovering new first-order properties via intrinsically-based criteria for the choice of preferred universes. It is open-ended, meaning that one cannot exclude the possibility of arriving at the statement you express above or even at the existence of LCs in V at some point in the future. But so far the evidence is just not there.

2) I challenge HP to establish that there is an inner model of “ZFC + there are infinitely many Woodin cardinals” without establishing PD. For this HP can use Con LC for any LC up to Axiom I0.

I think I understand the point you want to make here, which is that the HP so far offers no new techniques for producing consistency lower bounds beyond core model theory. I agree. But the intuitive (not mathematical) step from Con LC to inner models for LC is straightforward: Extrinsically we understand that LC is perfectly compatible with both the well-foundedness of the membership relation and with ordinal-maximality. From this we can conclude that LCs exist in countable transitive models of ZFC which are ordinal-maximal (i.e. #-generated). From this it provably follows that they exist in inner models.

We completely disagree on the Con LC issue as our email thread to this point clearly shows, no need to make a further comment on that.

You made 2 points. On point 2 we gentlemenly agree to disagree. But I may agree with you on point 1! Here is the challenge: Strengthen my argument that Con LC yields inner models for large cardinals to get inner models with LC’s containing any set (or even better, to get LC existence). Is there an intrinsically-based criterion (in the sense of my 19.August e-mail to Pen, item 5) that ensures this?

I agree that HP is part of the very interesting study of models of ZFC. There are many open and studied questions here. For example suppose $\phi$ is a sentence such that there is an uncountable wellfounded model of ZFC + $\phi$ but only at most one model of any given ordinal height. Must all the uncountable wellfounded models of ZFC + $\phi$ satisfy V = L? (The wellfounded models must all satisfy V = HOD and that there are no measurable cardinals). The answer could well be yes and the proof extremely difficult etc., but to me this would be no evidence that V = L.

See my 2012 MALOA lectures. A positive answer to your question follows from item 13 there; the proof is not difficult.

In my MALOA lectures I begin with a survey of purely mathematical properties of the Hyperuniverse and then specialise to those which are relevant to the HP. The HP is not part of the mathematical study of the Hyperuniverse, but rather mixes that study with philosophical considerations to develop a theory of intrinsically-based truth.

I don’t see what relevance your mathematical question has to the current discussion. But maybe you are proposing another criterion reflecting some intrinsic feature of the universe of sets (in the sense of my 19.August e-mail to Pen, item 3) compatible with “maximality”.

The issue I seek clarified is exactly how HP will lead to a new axiom. At some point HP must declare some new sentence as “true” . What are the HP protocols? You seem to suggest that SIMH if consistent is such a “truth” but I am not even sure you make that declaration.

This is clarified in my 19.August e-mail to Pen, item 7.

You said:

Let IMH(card-arith) be IMH together with the following:

Suppose there is a card-arith preserving extension of M in which \phi holds. Then there is an card-arith preserving inner model of M in which \phi holds.

“conjecture” : IMH(card-arith) implies GCH.

My question for Sy’s paper is simply, why if “conjecture” is true does one reject this in favor of SIMH (assuming SIMH is consistent)?

It is good that you posed this question because it illustrates very well how the HP is meant to work. If the “conjecture” is true then the SIMH is almost surely inconsistent and this would be exciting progress in the HP. Indeed I welcome the exploration of a wide range of such criteria in order to gain a better understanding of absoluteness, constantly refining our picture of the universe based on these forms of maximality. Of course some criteria, like the SIMH, are very natural and well-motivated, whereas others, such as the IMH for ccc extensions (roughly speaking: Levy absoluteness with cardinal-absolute parameters for ccc extensions) are not. Note that the latter is consistent and solves the continuum problem! But in my view its downfall is simply that the notion of “ccc extension” is unmotivated.

Continuing the point I make above. I agree with you that if “conjecture” is true then SIMH is probably inconsistent. But it is also possible that both “conjecture” is true and SIMH is consistent. What then?

This is the “bifurcation” issue. Again see item 7 of my 19.August e-mail to Pen.

I guess you could predict based on your conviction in HP that this latter case will not happen just as I predict PD is consistent. For me an inconsistency in PD is an extreme back-to-square-one event. I would like to see (at some point) HP make an analogous declaration.

Huh? The HP is a programme for truth in general, it is not aimed at a particular statement like CH. Even if the SIMH is inconsistent there is still plenty for the programme to explore. I don’t yet know if CH will have a constant truth value across the “preferred universes” (Sol is right, a better term would be something more flattering, like “optimal universes”), especially as a thorough investigation of the different intrinsically-based criteria has only just begun and it will take time to develop the optimal criteria together with their first-order consequences. I guess it is possible that an unavoidable “bifurcation” occurs, i.e. there are two conflicting optimal criteria, one implying CH and the other its negation. It is much too early to know that. Perhaps this would be what you call a “back-to-square-one event” regarding CH.

Best,
Sy