At a related thread at MSE an expert in reverse mathematics noted that "As the modern consensus is that only nonstandard models have infinitesimals, it will be quite challenging to give a concrete example of one." Such a challenge of giving concrete examples of infinitesimals would presumably also apply to infinite numbers.
I find this difficult to understand because if one has weak theories like Peano Arithmetic in mind, Skolem already constructed explicit models in 1933, in particular without relying on the axiom of choice. This was analyzed in detail by Stillwell in his article in the 1970s; see e.g., here.
Meanwhile, if one aims for the full power of the transfer principle as in Robinson's framework then an infinite integer $H$ would immediately produce nonmeasurable objects like $\{A\subseteq\mathbb{N}:H\in{}^{\ast}\!A\}$ so that a construction would be of difficulty comparable to Banach-Tarski and the like.
What is the nontrivial logical content of an assertion that infinite numbers are hard to come by, in the sense of reverse mathematics, perhaps in reference to theories of intermediate strength?
Best Answer
The difficulty involved in developing a theory of infinitesimals that would be useful in analysis is illustrated by the fact that, as discussed in the comments above, the surreals are unsuitable for this task. Classically one can produce nonstandard models of the integers in ZF, as discussed at related questions, but in a constructive context it does not seem particularly easy to prove a compactness theorem that would have similar ramifications. So perhaps a constructive context is an example of such "intermediate difficulty".