I agree with resolving the paradox along these lines: the judge was simply making a false statement.
Maybe a more satisfying way to reach that conclusion is to use Gödelian machinery. Interpret the judge’s statement S as saying “there is an integer N from 1 to 5, such that for any M from 1 to 5 the statement ‘N=M’ is not provable from the statement ‘N>M-1 and S is true’ and the axioms of PA”. Since the self-reference in S happens within a nested statement about provability, S can be interpreted as a statement about integers and arithmetic, using an arithmetized definition of provability in PA, and using the diagonal lemma (quining) to get its own Gödel number. And then yup, S can be shown to be false, by an argument similar to the paradox itself.
I guess I feel the need to not only show why the judge’s statement must be false if interpreted in the manner that the prisoner interpreted it, but to also show how non-contradictory interpretations of the judge’s statement manage to be almost, but not quite reflective.
I agree with resolving the paradox along these lines: the judge was simply making a false statement.
Maybe a more satisfying way to reach that conclusion is to use Gödelian machinery. Interpret the judge’s statement S as saying “there is an integer N from 1 to 5, such that for any M from 1 to 5 the statement ‘N=M’ is not provable from the statement ‘N>M-1 and S is true’ and the axioms of PA”. Since the self-reference in S happens within a nested statement about provability, S can be interpreted as a statement about integers and arithmetic, using an arithmetized definition of provability in PA, and using the diagonal lemma (quining) to get its own Gödel number. And then yup, S can be shown to be false, by an argument similar to the paradox itself.
I guess I feel the need to not only show why the judge’s statement must be false if interpreted in the manner that the prisoner interpreted it, but to also show how non-contradictory interpretations of the judge’s statement manage to be almost, but not quite reflective.