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I am exploring areas where non-standard analysis or the theory of surreal numbers has yielded results that remain exclusive to these frameworks without analogs or proofs in classical analysis. For example, Robinson's proof regarding polynomially compact linear operators on Hilbert spaces eventually found its translation into classical terms. Are there other significant theorems or results in non-standard analysis or surreal numbers that have not been similarly translated or reconciled with traditional methods in classical analysis?

While there is a transfer principle, I am curious if there are known results that explicitly rely on higher-order logic or other aspects outside the scope of the transfer principle? Or maybe even results in the first-order logic, but without standard proof, because as far as I understand the complexity of the proof can be exponentially increased. Examples or discussions on this topic would be greatly appreciated.

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    $\begingroup$ Here is a point of view on uniform continuity that seems to belong only to nonstandard analysis. It is not something that cannot be "reconciled" with standard methods, so I'm putting it in a comment rather than an answer. "Everybody knows" that a standard function $f$ is continuous if for every real number $a,$ the nonstandard counterpart of $f$ satisfies $f(a+\varepsilon)-f(a)$ is infinitesimal whenever $\varepsilon$ is infinitesimal. But if the same is true of every nonstandard real number $a$ rather than only of every real number $a,$ then $f$ is uniformly continuous. And conversely. $\endgroup$
    – Michael Hardy
    Commented Apr 17 at 16:45
  • $\begingroup$ For example $e^{a+\varepsilon} - e^a$ is infinitesimal if $\varepsilon$ is infinitesimal and $a$ is real. But if $a$ is an infinitely large nonstandard real, then for some infinitesimals $\varepsilon,$ that quantity is not infinitesimal. Similarly, if $a\ne0$ is infinitesimal then for some such $\varepsilon,$ $\sin(1/(a+\varepsilon)) - \sin(1/a)$ is not infinitesimal. $\endgroup$
    – Michael Hardy
    Commented Apr 17 at 16:48
  • $\begingroup$ One might think that if two assignments of probabilities to $0,1,2,3,\ldots$ summing to $1$ differ by a nonzero infinitesimal for every such integer, then their moments likewise differ by nonzero infinitesimals. But it can happen that their $n\text{th}$ moments for every $n \in \{1,\ldots, N\}$ for some infinite integer $N$ agree exactly, and for $n>N$ differ by more than an infinitesimal. That is what happens with the $\operatorname{Poisson}(1)$ distribution and the distribution of the number of fixed points of a uniformly distributed random permutation of a set of cardinality $N. \qquad$ $\endgroup$
    – Michael Hardy
    Commented Apr 17 at 16:54
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    $\begingroup$ I don't think that nonstandard analysis and surreal numbers should be mentioned in the same breath here, as if somehow, surreal numbers are highly relevant to nonstandard analysis. As explained elsewhere on MO, the surreal numbers are not particularly relevant to nonstandard analysis. $\endgroup$
    – Timothy Chow
    Commented Apr 17 at 18:37
  • $\begingroup$ @TimothyChow: moreover, they are not that relevant to analysis, either. $\endgroup$
    – Mikhail Katz
    Commented May 5 at 7:12

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The key paper in this area is Henson and Keisler:

C. W. Henson and H. J. Keisler, On the strength of nonstandard analysis}, J. Symbolic Logic, 51 (1986), no. 2, 377-386.

The elaborate on the point you alluded to. Namely, while nonstandard analysis is conservative over ZFC so in principle "no new results" appear, in practice using the nonstandard approach reduces the level of the logic by 1, so for example the nonstandard approach with second-order logic would be equivalent to the non-infinitesimal approach with third-order logic (which as you point out may be much harder to follow).

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