>> …the independence of a set-theoretic assertion from ZFC tells us little about whether it holds or not in the universe.
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> How can this be? If it is independent, then it cannot be proved from the axioms. Thus, one has the freedom to assume it or assume the negation, as an axiom. Why would someone expect to know “whether it holds or not in the universe”, if it has been proven independent? Wouldn’t that answer the question?
Hamkins' point is that it answers the question *if* you take a multiverse philosophical approach to set theory. However, if you believe that there is One True Set Theory it changes the question from "Does ZFC prove CH or ¬CH" to "Is ZFC+CH or ZFC+¬CH the correct choice of axioms?"
Peter Taylor
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2024-02-12T09:17:56Z (10 months ago)