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In "The set-theoretic multiverse," Hamkins talks about forcing giving us "glimpses" of other set-theoretic universes. He states his position as a Platonistic one, i.e. these "glimpses" are functionally equivalent to "perceptions" of actual objects.

In "The modal logic of forcing,", Hamkins (and Loewe) effectively claim that "is forceable" can mean "is possible." Now it seems that we can also say that "is forceable" can mean "is justifiable," for Hamkins, if we assume that "perceptions" of the states of affairs in a set-theoretic universe count as justifications of propositions about the contents of that universe. E.g. if we can force 21 = ℵ29, then we are justified in proposing that there is a universe where 21 = ℵ29.

My question is as follows. Let "JA" mean "it is justified that A" and "A →F B" mean "A is forced to B." (We assume that what is forced is an answer to a question, e.g., "CH or ~CH?" Otherwise we would say something like, "CH is forced to ~CH," which sounds wrong.) Now, on my reconstruction of Hamkins' viewpoint, does it make sense to say that we not only force sentences like, "A = B," but, "J(A = B)"? Moreover, then, what about, "J(A →F B)"? I.e. can forcings themselves be justified?†

One might say that a forcing sentence is justified if it "works." But Hamkins says that even granting a plurality of worlds, we do not necessarily grant that these worlds are on a par. The unique value of CH might be that it requires the minimum justification out of all proposed values for the Continuum. It is forcing-possible and forcing-justifiable; but consider then, "J(A →F J(CH)," i.e. is it justifiable to force the sentence, "CH is justified"? If so, if uniquely so—if this sentence, which requires minimal justification, is then maximally justified!—have we gotten an answer to CH while keeping Hamkins' multiverse, too?

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  • I don't understand your distinction between justifiability and forceability (especially given your second paragraph). But note that "forcing is transitive:" if V[G][H] is a generic extension of V[G] which is a generic extension of V, then V[G][H] is a generic extension of V (we have V[G][H]=V[GxH], and GxH will be generic over V for an appropriate iterated forcing). So "you can force the forceability of P" is equivalent to "P is forceable." Commented May 7, 2021 at 18:59
  • The interesting "multiple modality" statements are those which mix forceability with non-forceability: e.g. "there is a forcing extension W of V which has no forcing extension satisfying P." Some choices of P do give rise to principles of this form (e.g. "There is a real which is Cohen-generic over L"), while others don't (e.g. either of CH or ~CH). Commented May 7, 2021 at 19:00
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    Please, Hawkins is not a philosopher nor is he recognized as such. Commented May 9, 2021 at 4:23

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