Tag Archives: V as syntactic

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,
Radek