Modal set-theory

Question

In his “The Potential Hierarchy of Sets”, Review of Symbolic Logic 6:2 (2013), 205-28 Øystein Linnebo has proposed a modal set-theory. I was wondering what kind of utility can such a theory have for a mathematician and if there have been other similar attempts to develop theories of that kind.

Answer 1

Not an answer, but too long for a comment:

Something you may be interested in is the modal logic approach to forcing, which is probably the most important and ubiquitous technique in modern set theory, as developed in the book by Fitting and Smullyan (see the review http://www.jstor.org/stable/2586777?seq=2#page_scan_tab_contents, which includes a critique of the approach).

NOTE: by "modal logic approach to forcing" I mean the use of modal logic in providing a framework for forcing constructions, not the modal logics which arise in studying forcing extensions - as in http://arxiv.org/abs/math/0509616 - which is really cool but not the same thing.)

Of course, a caveat is in order here: how is the modal approach to forcing useful? Philosophically I am told it is appealing and intuitive, at least to some; mathematically, I don't know of any time it has been used to achieve new results. Much more useful is the approach via Boolean-valued models (see "A proof of the independence of the continuum hypothesis" http://link.springer.com/article/10.1007%2FBF01705520 by Scott for a gentle example, or the section in Jech's book for a more thorough approach) - this is similar in spirit, in that it uses non-classical logic and structures to achieve a purely syntactic approach to forcing. The advantage that Boolean-valued models have in forcing seems to be that the relevant combinatorial properties of Boolean algebras are well-understood and easy to work with; there are many results about forcing which are proved straightforwardly via Boolean-valued models, but whose proofs in the usual poset formulation are quite messy.

Answer 2

You should remember that for ZFC (and for ZF) the Power Set axiom doesn't say that 'the model has all subsets of a set S and they are collected into a single set'; rather, it only says that 'there is a set containing all subsets of a set S in the model'. Since each model of ZFC (ZF) is a possible world, a modal ZFC (ZF), say, would allow one to collect all possible subsets of a set S (from every model of ZFC (ZF) in which S exists) into a single set. This is, perhaps, more in line with the naive concept of 'set of all subsets' of a given set which the Power Set axiom was trying to express (which perhaps is the way ordinary mathematicians see the power set). See also Linnebo's "Grounded Abstraction" notes, which can be found under title on the Web. Note the distinctions between "Actualist generality" and "Potentialist generality" found on pg 5 of Linnebo's notes--this is an important (at least to me) distinction in the philosophy of set theory

Also, in order to help you to determine whether a modal set theory would be useful for mathematicians (i.e. ordinary mathematicians) take a look at Linnebo's paper "Pluralities and Sets" (you can find a copy of it on his homepage). I am currently reading through this paper now and am finding it very helpful.