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cousin_it comments on Second order logic, in first order set-theory: what gives? - Less Wrong
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If you believe that there's a unique standard model of the reals, you must also believe that the continuum hypothesis has a definite truth value. Some people don't believe that.
I don't think that's true. You may not believe that the set of functions is unique (in which case the notion of sets in bijection is no longer unique).
What exactly are you denying when you deny that the continuum hypothesis has a definite truth value? After all it's very easy to prove "CH is either true or false" in whatever formal system you prefer, with some notable but unpopular exceptions.
I'm not completely sure of that myself, but consider this analogy. Let PA+X be a formal system that consists of the axioms of PA plus a new axiom that introduces a new symbol X and simply says "X is an integer", without saying anything more about X. Then it's easy to prove "X is either even or odd" in PA+X, but it would be wrong to say that PA+X has a unique distinguished "standard model" that pins down the parity of X. So my statement about CH is more of a statement about our intuitions possibly misfiring when they say a formal system must have a unique standard model.
Are you comfortable rejecting the idea that PA has a "standard model"?