This will be my talk for the CUNY Graduate Center Philosophy Colloquium on November 28, 2012.

I will be speaking on topics from some of my recent articles:

- The set-theoretic multiverse
- The multiverse perspective on the axiom of constructibility
- Is the dream solution of the continuum hypothesis possible to achieve?

I shall give a summary account of some current issues in the philosophy of set theory, specifically, the debate on pluralism and the question of the determinateness of set-theoretical and mathematical truth. The traditional Platonist view in set theory, what I call the universe view, holds that there is an absolute background concept of set and a corresponding absolute background set-theoretic universe in which every set-theoretic assertion has a final, definitive truth value. What I would like to do is to tease apart two often-blurred aspects of this perspective, namely, to separate the claim that the set-theoretic universe has a real mathematical existence from the claim that it is unique. A competing view, which I call the multiverse view, accepts the former claim and rejects the latter, by holding that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe, and a corresponding pluralism of set-theoretic truths. After framing the dispute, I shall argue that the multiverse position explains our experience with the enormous diversity of set-theoretic possibility, a phenomenon that is one of the central set-theoretic discoveries of the past fifty years and one which challenges the universe view. In particular, I shall argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.

A couple of questions regarding your central theses: i) regarding your thesis that the claim that the set-theoretic universe has a real mathematical existence is distinct from the claim that the set-theoretic universe is unique, is the actual problematic claim (as Skolem correctly asserted –see his “Some remarks on axiomatized set theory” in van Heijenoort, pp 290-301) that ” when founded in such an axiomatic way, set theory cannot remain a privileged logical theory; it is then placed on the same level as as other axiomatic theories” (Skolem in van Heijenoort, pg 292)?

ii) when you assert that “the universe view holds that there is an absolute background concept of set and a corresponding absolute background set- theoretic universe in which every set-theoretic assertion has a final, definitive truth-value (either 0 or 1?), how do you interpret the boolean values of assertions in a boolean-valued model of ZFC which are not 0 or 1 ?

Thanks for your comment. I find the Skolem quotation you mention to be right on point. The universe view appears to offer a solution to the ontological problem, by explaining that to exist in mathematics means to be in the unique background universe of sets, but then suffers from the epistemological problem: how can we come to learn the fundamental truths of set theory? Many set theorists with the universe view fall back on a version of consequentialism, by taking the attractive consequences of their strongest theories, such as large cardinals, as evidence for the truth of those axioms. Meanwhile, in (ii), I don’t think there is an issue here. The universists simply view those Boolean values as mathematical calculations, not reflecting on actual truth or even possible truth, but rather as reflecting truth in some kind of virtual world.

Actually, Kunen in his text on set theory and independence proofs gives a nice interpretation of boolean values : ‘0’, if the sentence in question in the language of set theory holds in no B-extension (where B in a complete boolean algebra), ‘1’ if it holds in all B-valued extensions, and ‘b’, 0<b<1 if it holds in some but not all B-valued extensions. As in the frequency interpretation of probability, all M-people calculate the same 'frequencies' (though boolean values are obviously more general than probabilities). Since all models 'exist' (even though some may be considered 'imaginary') it might be said that the boolean-valued universe is actually the 'real' universe and therefore every mathematical statement has a definite boolean truth-value.

The phrase “B-valued extension should just be “B-extension”. Sorry. Also, it should read “(B is a complete boolean algebra).

Kunen calls them “B-generic extensions” but I wonder, can one use Boolean-valued models of ZFC to ‘construct’ ZFC’s inner models as well? If so, then one could drop the “generic” in B-generic models and say ||s|| =1 (s is a sentence in the language of set theory) iff s is true in all models of ZFC, ||s|| =0 iff s is true in no model of ZFC , and

0<||s||<1 iff s is true in some but not all models of ZFC, which would lend credence to the claim that the Boolean-valued universe is the 'real' universe and that therefore every mathematical statement has a definite Boolean truth-value.

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