Re: Paper and slides on indefiniteness of CH

Dear Harvey (and all other who follow the thread),

I will try to respond to your comments, hopefully providing some more clarification.

1] You say that it is crucial to answer the following:

Do we or do we not want to take the structure of ctms as somehow reflecting on the structure of the actual set theoretic universe?” It is important regarding the legitimacy of the use of ctm’s in HP.

Answer: For me, “the actual set-theoretical universe” (= V) is a meaningful term only if it refers to syntactical consequences of ZFC plus some other, explicitly assumed axioms. I do have an intuition about sets which might go beyond this, but it is just that – intuition, not expressible in words with any reasonable degree of accuracy. Starting with this modest assumption about V, it feels natural to work with models of ZFC and look at their properties (the usual “double” role of set theory — metatheory, and theory). The basic idea of HP, which I like, is that perhaps we learn more about our intuition by working with these models, providing we ask the right questions. Since V is either a set of provable sentences, or a vague subjective notion, the question how ctm’s correspond to V is off the target – ctm’s form a reasonably large collection of models, rich enough to provide a field for answering our questions (we decide at the beginning that ill-founded models and large models do not add more significant benefits; we choose transitive models = standard models, to have the standard numbers, formulas, etc).

This evidently does not answer your worries because you do think, as you wrote, that “But there is the real possibility of saying something generally understandable, surprising, and robust [about intrinsic maximality of sets], and therefore probably about V as well. I guess I am less optimistic, and therefore acknowledge that there will always be — at the beginning of the analysis — some “technical convenience”, it is just the question which convenience you prefer.

2] The question of whether AC follows from our intuition about sets. You asked,

Then what is all this talk on the traffic doubting whether AxC is supported by “intrinsic maximality of the set theoretic universe?

Answer: It seems to me that there is so much discussion regarding AC because people hope there is some “hidden proof” of AC from IMST (intrinsic maximality of set-theoretic universe), or some such similar notion. I do not share this hope myself – for the reason that the assumptions of IMST are too subjective to give rise to a widely acceptable argument (while I consider it probable there is some hidden clever proof of Fermat’s theorem, for instance — because here we have objective assumptions).

Aside. I confess i do not quite understand the meaning of “maximal iterative concept of sets = MIC”, either (MIC is sometimes used to argue for axioms of ZF+AC). Or rather, I understand the term MIC if it means an application of a transfinite recursion theorem as provable in ZF, in some maximal sense; I do not see how it can be used to argue for the axioms of ZF+AC (ordinals were defined by Cantor in set theory precisely to make proper sense of (transfinite) iteration, not conversely).

3] Finally, there is the name HP (and related vocabulary), to which you strongly object.

Answer: To me, the name should indicate family-resemblance to “multiverse” (which is open to similar discussion regarding its appropriateness). But let us for a moment forget about the name: would the project seem more convincing with a different name? If yes, I suggest the discussion continues while ignoring the current name; if no, the same applies. Let us not be distracted by the choice of vocabulary.

Best regards to all,

Re: Paper and slides on indefiniteness of CH

Dear all,

it seems that despite efforts of Sy, and some others, the same questions are raised over and over again. Recently, Harvey asked explicitly those who think that “HP is a legitimate foundations program” to write. I have collaborated with Sy on some of the mathematical papers regarding HP, and was a coauthor of one of the philosophical one.

I will attempt to answer briefly the questions posted by Harvey. My view on HP is different from Sy’s, but I see HP as a legitimate foundational program.

0] At a fundamental level, what does “(intrinsic) maximality in set theory” mean in the first place?

Let me write IMST instead of “(intrinsic) maximality in set theory” for the sake of brevity.

I doubt IMST can mean more than “viewing sets as big as possible, without the use of considerations based on practice of set theory as the main incentive”. “Intrinsic” is thus temporarily reduced to “non-extrinsic”; in view of the heavy philosophical discussions around this notion, I prefer to give it this more restrictive meaning. Note that “extrinsic”, unlike “intrinsic”, has a well-defined inter-subjective meaning. This leaves us with the word “big”; I guess that this is the primitive term, which cannot be defined by anything more simple — at least on the level of general discussion.

Admittedly, this definition is far from informative. For me, HP is a way of explicating this definition in a mathematical framework. Making its meaning more precise, and by the same token, less general. A discussion should be if other approaches — which set out to get real mathematical results — retain more of the general meaning of the term IMST. No approach can retain all the meaning of ISMT because it is by definition vague and subjective; thus HP should not be expected to do that.

1] Why doesn’t HP carry the obvious name CTMP = countable transitive model program.

Because the program was formulated by Sy with the aim of having wider application than the study of ctm’s.

2] What does the choice of a countable transitive model have to do with “(intrinsic) maximality in set theory”?

Countable models are a way of explicating IMST. It is a technical convenience which allows us to use model-theoretic techniques, not available for higher cardinalities.

Aside: I do not quite understand why the discussion rests so heavily on this issue: everyone seems to accept it readily when we talk about forcing (I know it can be eleminated in forcing, but the intuition — see Cohen’s book — comes from countable models). Would it make a difference if the models had cardinality \omega_1, or \omega and \omega_1, or should they be proper classes etc? Larger cardinalities would introduce technical problems which are inessential for the aims of HP.

3] Which axioms of ZFC are motivated or associated with “(intrinsic) maximality in set theory”? And why? Which ones aren’t and why?

IMST by historical consensus includes at this moment ZFC. “Historical consensus” for me means that many people decided that the vague meaning of IMST extends to ZFC. I do think that this depends on time (take the example of AC). HP is a way to raise some new first-order sentences as candidates for this extension.

4] What is your preferred precise formulation of IMH? E.g., is it in terms of countable models?


5] What do you make of the fact that the IMH is inconsistent with even an inaccessible (if I remember correctly)? If this means that IMH needs to be redesigned, how does this reflect on whether CTMP = HP is really being properly motivated by “(intrinsic) maximality in set theory”?

I view the process of obtaining results in HP like an experiment in explicating the vague meaning of IMST. It is to be expected that some of the results will be surprising, and will require interpretation.

6] What is the simplest essence of the ideas surrounding “fixing or redesigning IMH”? Please, in generally understandable terms here, so that people can get to the essence of the matter, and not have it clouded by technicalities.

It is a creative process: explicate IMST by principle P_1 — after some mathematical work, it outputs varphi (such as P_1 = IMH, \varphi = no inaccessible). Then try P_2, etc (P_2 can be a “redesigned”, or “modified” version of P_1). Of course, one hopes that his/her understanding of set theory will be helpful in identifying P‘s which have potential to output nice (good, deep) mathematics. It is essential that the principles P‘s should be as practice-independent as possible (= intrinsic, in my reading); that is what makes the program foundational (again, in my more narrow sense).

Best regards to all,
Radek Honzik