Dear Penny,

Many thanks for your insightful comments. Please see my responses below.

On Tue, 5 Aug 2014, Penelope Maddy wrote:

Thank you for the plug, Sol. Sy says some interesting things in his BSL paper about ‘true in V': it doesn’t ‘reflect an ontological state of affairs concerning the universe of all sets as a reality to which existence can be ascribed independently of set-theoretic practice’, but rather ‘a façon de parler that only conveys information about set-theorists’ epistemic attitudes, as a description of the status that certain statements have or are expected to have in set-theorist’s eyes’ (p. 80). There is ‘no “external” constraint … to which one must be faithful’, only ‘justifiable procedures’ (p. 80); V is ‘a product of our own, progressively developing along with the advances of set theory’ (p. 93). This sounds more or less congenial to my Arealist (a non-platonist): in the course of doing set theory, when we adopt an axiom or prove a theorem from axioms we accept, we say it’s ‘true in V’, and the Arealist will say this along with the realist; the philosophical debate is about what we say when we’re describing set-theoretic activity itself, and here the Arealist denies (and the realist asserts) that it’s out to discover the truth about some objectively existing abstracta. (By the way, I don’t think ‘truth-value realism’ is the way to go here. In its usual form, it avoids abstract entities, but there remains an external fact-of-the-matter quite independent of the practice to which we’re supposed to be faithful.)

My apologies here. In my reply to Sol I only made reference to truth-value realism for the purpose of illustrating that one can ascribe meaning to set-theoretic truth without being a platonist. Indeed my view of truth is very far from the truth-value realist, it is entirely epistemic in nature.

Unfortunately the rest of my story of the Arealist as it stands won’t be much help because the non-platonistic grounds given there in favor of embracing various set-theoretic methods or principles are fundamentally extrinsic and Sy is out to find a new kind of intrinsic support.

Yes. I am trying to make the case that there are unexplored intrinsic sources of evidence in set theory. Some have argued that we must rely solely on extrinsic sources, evidence emanating directly from current set-theoretic practice, because intrinsic evidence cannot take us past what is derivable from the maximal iterative conception. I do agree that this conception can lead us no further than reflection principles compatible with V = L.

But in fact my intuition goes further and suggests that no intrinsic first-order property of the universe of sets will enable us to resolve problems like CH. We have to examine features of the universe of sets that are only revealed by comparing it to other possible universes (goodbye Platonism) and infer first-order properties from these “higher-order” intrinsic features of V (a name for the epistemically-conceived universe of sets).

Obviously a direct comparison of V with other universes is not possible (V contains all sets) so we must instead content ourselves with the comparison of pictures of V. These pictures are perfectly provided by the hyperuniverse (also conceived of non-platonistically). And by Löwenheim-Skolem we lose none of the first-order features of V when we model it within the hyperuniverse.

Now consider the effect that this has on the principle of maximality. Whereas the maximal iterative concept allows us to talk about generating sets inside V by iterating powerset “as long as possible”, the hyperuniverse allows us to express the maximality of (a picture of) V in a more powerful way: maximal means “as large as possible in comparison to other universes” and the hyperuniverse gives a precise meaning to this by providing those “other universes”. Maximality is no longer just an internal matter regarding the existence of sets within V, but is also an external matter regarding the largeness of the universe of sets as a whole in comparison to other universes. Thus the move from the concept of set to the concept of set-theoretic universe.

Now comes a crucial point. I assert that maximality is an intrinsic feature of the universe of sets. Certainly I can assert that there is a rich discussion of maximality in the philosophy of set theory literature with some strong advocates of the principle, including Goedel, Scott and yourself (correct me if I am wrong).

Maximality is not the only philosophical principle regarding the set-theoretic universe that drives the HP but surely it is currently the most important one. Another is omniscience (the definability in V of truth across universes external to V). Maybe there will be more.

I’m probably insufficiently attentive, or just plain dim, but I confess to being confused about how this new intrinsic evidence is intended to work. It isn’t a matter of being part of the concept of set, nor is it given by the clear light of mathematical intuition. It does involve, quoting from Gödel, ‘a more profound understanding of basic concepts underlying logic and mathematics’, and in particular, in Sy’s words, ‘a logical-mathematical analysis of the hyperuniverse’ (p. 79). Is it just a matter of switching from the concept of set to the concept of the hyperuniverse? (My guess is no.) Our examination of the hyperuniverse is supposed to ‘evoke’ (p. 79) certain general principles (the principles are ‘based on’ general features of the hyperuniverse (p. 87)), which will in turn ‘suggest’ (pp. 79, 87) criteria for singling out the preferred universes — and the items ultimately supported by these considerations are the first-order statements true in all preferred universes. One such general principle is maximality, but I’d like to understand better how it arises intrinsically out of our contemplation of the hyperuniverse (at the top of p. 88). On p. 93, the principle (or its more specific versions) is said to be ‘the rigorous expression of what it means for an element of the hyperuniverse, i.e., a countable transitive model of ZFC, to display “maximal properties”‘. Does this mean that maximality for the hyperuniverse derives from a prior principle of maximality inherent in the concept of set?

You ask poignant questions; I hope that what I say above is persuasive!

Many thanks for your interest, and very best wishes,

Sy