Hyperreal infinitesimal probabilities and definability

In order to assign non-zero probabilities to such things as a lottery ticket in an infinite fair lottery or hitting a specific point on a target with a uniformly distributed dart throw, some people have proposed using non-zero infinitesimal probabilities in a hyperreal field. Hajek and Easwaran criticized this on the grounds that we cannot mathematically specify a specific hyperreal field for the infinitesimal probability. If that were right, then if there are hyperreal infinitesimal probabilities for such a situation, nonetheless we would not be able to say what they are. But it’s not quite right: there is a hyperreal field that is "definable", or fully specifiable in the language of ZFC set theory.

However, for Hajek-Easwaran argument against hyperreal infinitesimal probabilities to work, we don’t need that the hyperreal field be non-definable. All we need is that the pair (^*R,α) be non-definable, where ^*R is a hyperreal field and α is the non-zero infinitesimal assigned to something specific (say, a single ticket or the center of the target).

But here is a fun fact, much of the proof of which comes from some remarks that Michael Nielsen sent me:

Theorem: Assume ZFC is consistent. Then ZFC is consistent with there not being any definable pair (^*R,α) where ^*R is a hyperreal field and α is a non-zero infinitesimal in that field.

[Proof: Solovay showed there is a model of ZFC where every definable set is measurable. But every free ultrafilter on the powerset of the naturals is nonmeasurable. However, an infinite integer in a hyperreal field defines a free ultrafilter on the naturals—given an infinite integer M, say that a subset A of the naturals is a member of the ultrafilter iff |M| ∈ ^*A. And a non-zero infinitesimal defines an infinite integer—say, as the floor of its reciprocal.]

Given the Theorem, without going beyond ZFC, we cannot count on being able to define a specific hyperreal non-zero infinitesimal probability for situations like a ticket infinite lottery or hitting the center of a target. Thus, if a friend of hyperreal infinitesimal probabilities wants to be able to define one, they must go beyond ZFC (ZFC plus constructibility will do).