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Is everything that is logically possible also mathematically possible, and vice versa? Note, I am not suggesting that logic and mathematics are identical. I am merely asking whether logical possibility is the same thing as mathematical possibility. Even if logic and mathematics are not identical, it could be that logical possibility is coextensive with mathematical possibility.

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  • What is mathematical possibility? Commented Dec 6, 2022 at 5:55
  • Does this answer your question? Are "mathematically possible universes" the same as "logically possible universes"? Commented Dec 6, 2022 at 21:59
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    No, logical possibility is strictly broader. Mathematical possibility is defined relative to some background mathematical formalism, typically ZFC, which includes non-logical constraints (axioms). So, for example, it is logically possible that infinite sets do not exist, but mathematically it is impossible because their existence is postulated by the axiom of infinity.
    – Conifold
    Commented Dec 7, 2022 at 3:39
  • @Conifold You should probably put that as an answer.
    – user107952
    Commented Dec 7, 2022 at 4:57

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This is very vague due to the fact that little exposition has been done on the background modality in question (the way "logical modality" behaves is not all that clear: this depends on how one demarcates the problem of logical constants and various other issues stemming back as far as Carnap, Wittgenstein & Tarski).

Similarly, not very much detail has been given on what is meant by mathematics here. If we were to hypothetically say that "mathematics" just is set theory, like ZF, formulated through interpreting a first order theory like FOL, then there will be models of FOL that are not models of ZF, so you could argue that the "logical modality" is "broader".

I'm putting scare quotes around 'broader' because, strictly speaking, the amount of possible worlds is not expressible in set theory, and so we cannot say things like "the set of logically possible worlds is larger than the set of mathematically possible worlds" (because neither of these are sets: this is well known although some outdated philosophy lags behind, see Possible Worlds & The Upper Bound Problem)

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  • In your quoted ref the SPACETIME of arbitrary cardinality is used to refute bounded set-theoretic L-ersatzism which seems a more generic problem facing empirical physics theories too which are all based on ZF set theories at least implicitly? Commented Dec 7, 2022 at 21:39

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