On Sun, 3 Aug 2014, Solomon Feferman wrote:
Thanks for your helpful comments on my draft, “The Continuum Hypothesis is neither a definite mathematical problem nor a definite logical problem,” and especially for bringing your Hyperuniverse Program (HP) to my attention. I had seen your 2013 article with Arrigoni on HP back then but had not taken in its point. I have now read it as well as your Chiemsee slides, and will certainly take it into account in the final version of my paper. I’m glad that we are in considerable agreement about my fundamental argument that one must distinguish mathematical problems in the ordinary sense from logical problems, and that as of now what I claim in the title is true, even taking HP into consideration. Is my title misleading since it does not say “as of the time of writing”? The reader will see right away in the abstract and the opening section that what I claim does not exclude the possibility that in the future CH will return as a definite mathematical problem [quite unlikely] or that it will somehow become a definite logical problem.
This does appear to constitute a significant retreat in your position. In the quote of yours that I used in my Chiemsee tutorial you refer to CH as being “inherently vague”, in other words dealing with concepts that render it impossible to ever assign it a truth value. If you now concede the possibility that new ideas such as hinted at by Gödel in the quote below (and perhaps provided by the hyperuniverse programme) may indeed lead to a solution, then the “inherent vagueness” argument disappears and our positions are quite close. Indeed we may only differ in the degree of optimism we have about the chances of resolving ZFC-undecidable problems in abstract set theory through philosophically-justifiable logical methods.
“(Gödel) Probably there exist other axioms based on hitherto unknown principles … which a more profound understanding of the concepts underlying logic and mathematics would enable us to recognize as implied by these concepts.”
This is not the place to respond to your many interesting comments on the draft, nor on the substance of the HP and your subsequent exchange with Woodin. But I would like to make some suggestions regarding your terminology for HP (friendly to my mind). First all, it seems to me that “preferred models” is too weak to express what you are after. How about, “premier models” or some such? (Tapping into the Thesaurus could lead to the best choice.)
I do see your point here, because I do want to suggest not simply a “preference” for certain universes over others but rather a “compelling” or “justified” preference. I’ll give the terminology more thought, thanks for the comment.
Secondly, I’m not happy about your use of “intrinsic evidence for set-theoretic truth” both because “intrinsic evidence” is commonly used to refer to the constellation of Gödel’s ideas in that respect (not the line you are taking) as opposed to “extrinsic evidence”, and because “set-theoretic truth” suggests a platonistic view (which you explicitly reject). I don’t have anything to take its place, but it reminds me of the kinds of methodological maxims that Maddy has promoted, so perhaps a better choice of terminology can be found in her writings in place of that.
I do not think that “set-theoretic truth” entails a platonistic viewpoint (indeed there is a concept of “truth-value determinism” that falls short of Platonism). The goal of the programme is indeed to make progress in our understanding of truth in set theory and a key claim is that there is intrinsic evidence regarding the nature of the set-theoretic universe that transcends the older form of such evidence emanating from the maximal iterative conception. I think that the dichotomy intrinsic (a priori) versus extrinsic (a posteriori) which Peter Koellner has emphasized is a valuable way to clarify the debate. Nevertheless I do appreciate that some have suggested that the distinction is not as sharp as I may have assumed and I would like to hear more about that.
Another very interesting question concerns the relationship between truth and practice. It is perfectly possible to develop the mathematics of set theory without consideration of set-theoretic truth. Indeed Saharon has suggested that ZFC exhausts what we can say regarding truth but of course that does not force him to work just in ZFC. Conversely, the HP makes it clear that one can investigate truth in set theory quite independently from set-theoretic practice; indeed the IMH arose from such an investigation and some would argue that it conflicts with set-theoretic practice (as it denies the existence of inaccessibles). So what is the relationship between truth and practice? If there are compelling arguments that the continuum is large and measurable cardinals exist only in inner models but not in V will this or should this have an effect on the development of set theory? Conversely, should the very same compelling arguments be rejected because their consequences appear to be in conflict with current set-theoretic practice?
Best wishes and many thanks,