Can infinitely large positive integers in nonstandard analysis be identified with nonstandard natural numbers as described under Scott Aaronson's latest blog post? I.e. a certain Turing machine runs forever, but we can't prove it runs forever, so we can consistently assert that it halts. In this case the number of steps until it halts is a nonstandard natural number.
Yes, that is exactly right. Whenever a program does not halt, but this is not provable, it is precisely because there is some nonstandard model of arithmetic in which the program halts at a nonstandard stage of computation.
Yes, there is a transfer principle to the surreals (assuming global choice), since it is a saturated real-closed field. Just being a large ordered field is not enough, since one will want it to be real-closed, which means that positive elements have square roots and odd-degree polynomials have roots. In the plain language of fields, this implies that the field is definably complete (that is, every definable nonempty bounded set has a LUB), but when expanding the language one wants definable completeness for the transfer principle, since the reals have that property. Meanwhile, every saturated set-sized real-closed field extension of the reals will admit the transfer principle, and there are arbitrarily large instances of this in ZFC.
If by “the smallest” you mean size continuum, then you need to assume the continuum hypothesis. In fact, professor Hamkins argues that in an alternate history, where hyperreal analysis would have been formalized before epsilon delta calculus, we might have seen CH as a fundamental axiom.
Can infinitely large positive integers in nonstandard analysis be identified with nonstandard natural numbers as described under Scott Aaronson's latest blog post? I.e. a certain Turing machine runs forever, but we can't prove it runs forever, so we can consistently assert that it halts. In this case the number of steps until it halts is a nonstandard natural number.
Yes, that is exactly right. Whenever a program does not halt, but this is not provable, it is precisely because there is some nonstandard model of arithmetic in which the program halts at a nonstandard stage of computation.
"the hyperreal numbers” might reasonably be given meaning by taking it to refer to the surreal numbers"
Interesting. Usually the *smallest* instance of a type is the natural choice of representative.
Is there a transfer principle for the surreal numbers? Can any sufficiently large ordered field be endowed with a transfer principle?
Yes, there is a transfer principle to the surreals (assuming global choice), since it is a saturated real-closed field. Just being a large ordered field is not enough, since one will want it to be real-closed, which means that positive elements have square roots and odd-degree polynomials have roots. In the plain language of fields, this implies that the field is definably complete (that is, every definable nonempty bounded set has a LUB), but when expanding the language one wants definable completeness for the transfer principle, since the reals have that property. Meanwhile, every saturated set-sized real-closed field extension of the reals will admit the transfer principle, and there are arbitrarily large instances of this in ZFC.
If by “the smallest” you mean size continuum, then you need to assume the continuum hypothesis. In fact, professor Hamkins argues that in an alternate history, where hyperreal analysis would have been formalized before epsilon delta calculus, we might have seen CH as a fundamental axiom.
See https://jdh.hamkins.org/the-continuum-hypothesis-could-have-been-a-fundamental-axiom-oslo-june-2024/
Yes, indeed, and in fact the existence of a unique smallest-size countably saturated real-closed field is equivalent to CH, as I argue in the paper.