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disregardthat
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Hurkyl said:Statements don't have truth values, not even tautologies. "Truth" is a matter of semantics -- e.g. each set-theoretic interpretation yields a truth valuation: a function that maps the set of statements to the set {true, false}. (and a model of a theory is one in which each of its theorems map to true)
Interesting, it makes sense that only a set-theoretic interpretation determines the truth of any well-defined statement in set theory. But still, I have thought of the undecidability of CH as something different. I have thought of (the interpretation of) CH as a statement not bearing a truth value regardless of any set-theoretic interpretation (in which CH is interpreted), thus differentiating this "type" of undecidability from the pure syntactical statements, which are not bearing any truth value.
EDIT: By context I meant a set-theoretic interpretation.
Hurkyl said:CH is undecided in ZFC because it is neither provable nor disprovable. CH is, of course, provable in ZFC+CH.
For any particular (classical) model of ZFC, the truth value of CH is either true or false.
Oh, so it does make sense to call a statement provable if it doesn't bear a truth value. Using your definitions; the syntax determines the provability of a statement, but CH is not provable (in ZFC).
If so, how does it make sense to call CH semantically true (or false) in a set-theoretic interpretation of ZFC if it is syntactically unprovable (in ZFC)?
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