I was going to say “Congratulations, you just proved the halting theorem.”—but actually I think the paradox you’re gesturing at fails to “work” for shallower reasons (e.g., the reals not being well-ordered—trivially not by the usual ordering, and less trivially not by anything computable, because it’s consistent with ZF for there to be no well-ordering of the reals).
And among these, there is the smallest undefinable number.
There isn’t. The collection of positive, undefinable numbers is bounded below by zero, but doesn’t actually have a smallest element (due to the reals not being well-ordered).
And among these, there is the smallest undefinable number.
I was going to say “Congratulations, you just proved the halting theorem.”—but actually I think the paradox you’re gesturing at fails to “work” for shallower reasons (e.g., the reals not being well-ordered—trivially not by the usual ordering, and less trivially not by anything computable, because it’s consistent with ZF for there to be no well-ordering of the reals).
There isn’t. The collection of positive, undefinable numbers is bounded below by zero, but doesn’t actually have a smallest element (due to the reals not being well-ordered).
If you mean the smallest undefinable positive number, then isn’t that epsilon?