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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?

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    $\begingroup$ The motivation for the thread on math.SE seems to have been pedagogical: if one is teaching freshman calculus using an approach similar to Keisler's, how does one describe this issue to the students? In that context, I think the answer is clear. The students are learning a body of practices for manipulating infinitesimals, and these practices have been standardized and in use by scientists and engineers without interruption since Leibniz and Newton. In that body of standard practices, we never distinguish an individual infinitesimal. $\endgroup$
    – user21349
    Commented Jan 8, 2016 at 16:03
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    $\begingroup$ This question misunderstands Carl Mummert's remark, which is not saying much more than that there are no infinitesimals in the standard model of first-order PA. This fact does not imply any sweeping statement such as "infinite numbers are hard to come by" or that there is a "difficulty of giving concrete examples of infinitesimals." For example, in the context of surreal numbers (rather than nonstandard analysis), one can give concrete definitions of certain infinitesimals as being the values of certain finite games. The question should be rephrased to eliminate false presuppositions. $\endgroup$
    – Timothy Chow
    Commented Jan 8, 2016 at 16:53
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    $\begingroup$ katz, you should ask Carl Mummert to clarify his comment on MSE before asking this question. At the risk of additionally stirring muddy waters, there are no computable nonstandard models of PA: wikiwand.com/en/Tennenbaum's_theorem $\endgroup$
    – François G. Dorais
    Commented Jan 10, 2016 at 0:23
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    $\begingroup$ katz, the question is specifically about Carl Mummert's comment. He's in an ideal position to clarify what he meant. For what it's worth, my reading of Mummert's comment is similar to Tim's. Nothing is being said about the difficulty of finding non-standard models, it's only said that non-standard numbers may be hard to find since, according to common belief, one model has no non-standard numbers. $\endgroup$
    – François G. Dorais
    Commented Jan 10, 2016 at 15:12
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    $\begingroup$ katz, since you have yet to ask Carl Mummert to clarify his statement on MSE and you have yet to consider the possibility that you might have misunderstood what he wrote, there is no cause to reopen this question. $\endgroup$
    – François G. Dorais
    Commented Jan 10, 2016 at 17:28

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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".

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