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.) 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.
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?
With all best wishes,