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One subsequence in the argument I'm working on goes something like this:

  1. ♢A → □♢A.
  2. ¬□♢A.
  3. ∴ ¬♢A.

This seems valid (it's modus tollens, no?) but it seems to make the actual argument too "easy" (I worry there's an equivocation involved):

  1. ∄ℵω → ¬□♢(℘(ℵn) > ℵm)
  2. ♢(℘(ℵn) > ℵm) → □♢(℘(ℵn) > ℵm)
  3. ♢¬□♢(℘(ℵn) > ℵm)
  4. ∴ ¬♢(℘(ℵn) > ℵm)

(The assumption, "If A, then necessarily A," is only meant for possible mathematical truths. Obviously in general this assumption would be flawed. Even mathematically, it is tantamount to a rejection of the multiverse standpoint. For now, my only attempt to compensate for this weakness in the presentation of the argument is to try to limit it to, again, possible mathematical truth, if this is doable.†)

Wouldn't necessary possibility be possibility in all possible worlds? But then isn't that a different sense of "possible" than in the phrase "possible worlds"?

I also do not assume that possible necessity reduces to actual necessity, or even actuality as such. This reduction has been used in a poetic argument for the existence of God, but perhaps just for that reason it ends on a suspicious note. At any rate, I'm not even sure possible worlds semantics are what I'm using in the first place. The recent update to "Medieval Theories of Modality" seems to spell out part of my own theory of modal logic, viz. that assertions of possibility more or less reduce to complex conditional and disjunctive assertions.

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    You are not using "if A, then necessarily A", you are using "if possibly A, then necessarily possibly A". Applying "possibly" to A integrates out all dependence on possible worlds, any string of modal operators in front of "possibly" is then superfluous and can be removed. You may be using something weaker than S5 (where ♢A → □♢A is valid) or two dimensional modality, but how would we know what your theory is if even you are not sure?
    – Conifold
    Apr 21 at 0:38
  • It is not clear (to me) how you propose to interpret your modality in a mathematical context. Mathematical propositions are usually considered to be unimodal. They are either true or false, or if you prefer, necessarily true or necessarily false. An unproven conjecture could be considered to be epistemically possible, but that doesn't seem to be what you are aiming at expressing here. Without knowing what the intended semantics of the □ and ♢ operators are, it is difficult to know what to make of your theory...
    – Bumble
    Apr 21 at 14:05
  • ...Are you trying to set up a kind of meta-theory that ranges over different possible ways of doing mathematics? If so, I applaud your ambition, though we would perhaps need to see some more details.
    – Bumble
    Apr 21 at 14:05
  • If Bumble is right in what you are doing A=A(x,y), where x is a set theory ("possible way of doing mathematics") and y is its model. So your modal logic has two dimensions, one epistemic that ranges over (some class of) set theories, and the other metaphysical, that ranges over their models. Then ♢A = ∃y A(ZFC,y) (taking x=ZFC as our actual set theory), and □♢A = ∀x∃y A(x,y). Obviously, ♢A → □♢A is invalid. It is easier to dispense with modal operators and just use quantifiers explicitly.
    – Conifold
    Apr 21 at 15:29
  • I've had to distinguish between a set-theoretic multiverse resulting from forcing, and one resulting from different axioms, sort of a Tegmarkian gloss of the matter I guess. For example, correct me if I'm wrong, you can't force the existence of nonwell-founded sets in a system with the axiom of foundation, so a universe containing them would be a "parallel world" that wasn't a forcing extension of the WF form of V? Apr 21 at 21:05
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The general principle "possibility implies necessary possibility" is actually rarely true in the various modal set-theoretic conceptions I'm familiar with. For example, suppose we interpret "is possible" as "can be forced." If we start with L, then V=L is possible (since it's true already) but it's not necessarily possible (as soon as we apply any nontrivial forcing it breaks, and it can't be "brought back").

So right off the bat, before you can run the argument you intend you need to carefully pin down exactly what sort of "modal set theory" you're looking at. (This is related to my answer to an earlier question of yours.)

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  • "as soon as": this makes me think of a temporal form of V, an evolving universe of sets. Something being impossible "after" forcing: like the necessity of the past? Apr 21 at 22:38
  • @KristianBerry Well, this is basically the picture that (one of) the set theoretic multiverse(s) paints. Hamkins distinguishes between "buttons" (statements which, once forced, stay forced -you can't unpush a button) and "switches" (statements which can always be forced and whose negations can always be forced). For example, ~V=L is a button but CH is a switch. Apr 21 at 23:10

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