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?)


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,

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!

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