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2 fixed a typo

I think the following is implicit in earlier answers, but let me state it briefly. I'm addressing only the original question, not CH or other matters that gradually entered this discussion.

The main point is that a model of ZFC can have non-standard natural numbers; they're larger than all the standard ones, so we sometimes call them "infinite" even though from the point of view of the model they're finite. (That is, they satisfy, in the model, the formula that defines "finite ordinal number".) Now suppose such a model satisfies "There is PTime algorithm for SAT." Fix such an algorithm in the model, say a Turing machine program. It is true in the model that this algorithm has a finite number of control states (because that's part of the definition of "Turing machine") and that it's its running time is bounded by a polynomial, of some finite degree, of the input length (because that's part of the definition of "PTime"). Unfortunately, both of the occurrences of "finite" in the preceding sentence are (like the whole sentence) to be understood in the sense of the model. Neither the number of states nor the degree of the polynomial will necessarily be an actual natural number (in the sense of the real world rather than the model); they can be non-standard numbers. So the model's Turing machine might not be an actual Turing machine, and, even if it is, its running time might not be bounded by an actual polynomial.

1

I think the following is implicit in earlier answers, but let me state it briefly. I'm addressing only the original question, not CH or other matters that gradually entered this discussion.

The main point is that a model of ZFC can have non-standard natural numbers; they're larger than all the standard ones, so we sometimes call them "infinite" even though from the point of view of the model they're finite. (That is, they satisfy, in the model, the formula that defines "finite ordinal number".) Now suppose such a model satisfies "There is PTime algorithm for SAT." Fix such an algorithm in the model, say a Turing machine program. It is true in the model that this algorithm has a finite number of control states (because that's part of the definition of "Turing machine") and that it's running time is bounded by a polynomial, of some finite degree, of the input length (because that's part of the definition of "PTime"). Unfortunately, both of the occurrences of "finite" in the preceding sentence are (like the whole sentence) to be understood in the sense of the model. Neither the number of states nor the degree of the polynomial will necessarily be an actual natural number (in the sense of the real world rather than the model); they can be non-standard numbers. So the model's Turing machine might not be an actual Turing machine, and, even if it is, its running time might not be bounded by an actual polynomial.