- J. D. Hamkins, “The set-theoretic multiverse,” Review of Symbolic Logic, vol. 5, pp. 416-449, 2012.
`@ARTICLE{Hamkins2012:TheSet-TheoreticalMultiverse, AUTHOR = {Joel David Hamkins}, TITLE = {The set-theoretic multiverse}, JOURNAL = {Review of Symbolic Logic}, YEAR = {2012}, volume = {5}, number = {}, pages = {416--449}, month = {}, note = {}, url = {}, doi = {10.1017/S1755020311000359}, abstract = {}, keywords = {}, source = {}, eprint = {1108.4223}, archivePrefix = {arXiv}, primaryClass = {math.LO}, url = {http://jdh.hamkins.org/themultiverse}, }`

The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous diversity of set-theoretic possibilities, a phenomenon that challenges the universe view. In particular, I 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.

Multiversive at n-Category Cafe | Multiverse on Mathoverflow

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Very thought-provoking! I’m glad to see that you emphasized the ‘algebraic’ nature of ‘models of set theory’. For whatever it’s worth, I think that that’s proper. However, just because there exist exotic ‘subalgebras’ of V (deeming V to be a very generalized ‘algebra’) does not mean that the ‘exotic subalgebras’ constitute

the ‘proper subalgebras’ of V (I define a proper submodel of V to be any model–but no–taking your multiverse view how can one even come to an adequate conception of such, given that, according to you, the cumulative knowledge of the multiverse is necessary for a ‘true knowledge’ of ‘set’). Therein lies the problem. If a concept is perfectly elastic, what good is it?

It seems to me that the multitude of models of set theory, constructed (so to speak) by any one of a number of means has so stretched the meaning that

it has no coherent meaning at all, or at the very least, has no real usefulness for mathematics proper (for example, how would one go about defining a ‘generic’ Dedekind Cut?). Perhaps one could say that the models (of whatever kind) of ZFC, NF, Morse-Kelly, Godel-Bernays, etc. are simply the precise formulations of Cantor’s original definition of set (“By an ‘aggregate’ we are to understand any collection [of any kind whatsovever–my comment] into a whole M of definite and separate objects m of our intuition or our thought.”) and if that is the case, then certainly the multiverse view you espouse is unproblematic. But then which type of collection is to be adequate for the formulation of mathematics (how much classical mathematics could be formulated, say, in L)? What collection of models will now, in your view, be deemed as ‘standard’ (perhaps a synonym for proper)? What is to be the criteria?

I wish to correct a term that might make my previous comment somewhat confusing. Replace the term ‘proper’ in ‘proper subalgebra’, ‘proper model’, and ‘proper’ in the phrase “perhaps a synonym of proper” with the term ‘preferred’. The question then becomes, in essence, what are the criteria for distinguishing the preferred submodels of the set-theoretic universe from the exotic? Both the preferred submodels and the exotic submodels certainly exist, but only the preferrred submodels directly contribute to an understanding of the term ‘set’ and help determine its meaning. For my part, L would have to be the basis for constructing the preferred submodels of V as it is the minimal submodel of V (it would constitute, so to speak, the ‘skeletal structure’ of V) and the ‘preferred’ forcing extensions of L would have to add ‘all possible’ subsets A of a set X to P(X) just short of inconsistency so that the cardinal structure of L holds in all the preferred forcing extensions (that way the universal quantifier (A) in (For some B)(A)( If A is a subset of X then A is a member of B) would make P(X) absolute). Perhaps one could redefine P(X) via some inductive definition where the induction operator (call it P) could be applied to X to get more and more ‘new’ subsets of X in such a way that the closure ordinal would be an Aleph. Just speculation, of course, but a possible start ( in fact, one might characterize V as L plus forcing extensions since the ZFC -provable principle of forcing are exactly those in the modal theory S4.2 )….

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