Surreal numbers are constructed 'explicitly', many of them have labels on which one can do arithmetic, an extension of the natural ordinal arithmetic on Cantor's normal forms, and their theory is developed 'naively', keeping symbolic logic to a minimum. In non-standard analysis, on the other hand, infinitesimals and other hyperreals are treated as ghosts devoid of any individuality, and much attention is paid to logical formulas instead of numbers and functions themselves. This seems to be the main obstacle to its wider use. Why not give them labels based on representing sequences, do arithmetic on them, and draw down on "properties expressible in the first order logic" or headache inducing distinctions between internal and external sets? Especially since hyperreals can be embedded into surreals.

Why can't we fix some non-principal ultrafilter for the duration and work with hyperreals pretending as if they actually exist, like we do with reals and surreals? Is it a legacy difference in style going back to Conway and Robinson, or is there something deeper?

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    $\begingroup$ The surreals do not extend ordinal arithmetic. The surreal's arithmetic is a field's arithmetic, in particular commutative and cancellative; ordinal arithmetic is neither in the case of infinite ordinals. $\endgroup$ – Asaf Karagila May 29 '14 at 0:58
  • $\begingroup$ As for the question[s] in the last paragraph, an ultrafilter which we will fix will need to have at least some canonicity properties. Things like some sort of minimality. But we can often arrange for these properties of ultrafilter to be destroyed via forcing. So when changing models of $\sf ZFC$ we effectively change the surreal numbers in an unrecognizable way (where as the real numbers are always the completion of the rationals). Moreover, whereas we can imagine the real numbers do exist and our world is continuous, hyperreal include transfinite, so they immediately become intangible. $\endgroup$ – Asaf Karagila May 29 '14 at 1:02
  • $\begingroup$ Did you mean "hyperreal" after "effectively change"? And ordinals are also transfinite but we pretend they exist in the sense that naive expositions manipulate them without resorting to logical formulas. But even basic treatments of non-standard analysis talk about elementary equivalence, transfer principle, etc., which is like trying to create an illusion while showing how it's done. It seems that higher level of abstraction is used to explain infinitesimals compared to ordinals, what is the reason for such 'linguistic' approach? Is it essential in applications? $\endgroup$ – Conifold May 29 '14 at 8:26
  • $\begingroup$ I'm a bit puzzled by your seeming to express a contrast between "existing" and "being discussed with logical machinery". $\endgroup$ – Malice Vidrine May 29 '14 at 9:20
  • $\begingroup$ I find it much harder to explain things to people when object level and meta-level are present at the same time. Ordinals can be explained naively first, and when some experience is gained logical subtleties can be added. But in non-standard analysis both levels seem to be mixed from the start (transfer principle, etc.). There is a mental obstacle to working with infinitesimals when they are some extras that mysteriously fall under the scope of the same formula which 'normally' just produces $0$. I am talking about pretend existence for psychological reasons. $\endgroup$ – Conifold May 29 '14 at 10:05

The approach you recommend is basically the one adopted in Goldblatt's book:

Goldblatt, Robert. Lectures on the hyperreals. An introduction to nonstandard analysis. Graduate Texts in Mathematics, 188. Springer-Verlag, New York, 1998.

Note that the ultrapower construction of the hyperreals is already in a 1948 paper by E. Hewitt:

Hewitt, Edwin. Rings of real-valued continuous functions. I. Trans. Amer. Math. Soc. 64, (1948). 45–99.

As far as your question of "actual existence" of reals versus hyperreals is concerned, while it is true that each hyperreal that's not real is undefinable, it is also true that almost all real numbers are undefinable.

While the real number system itself is definable, it turned out to everybody's surprise that the same can be said for the hyperreal number system; see

Kanovei, Vladimir; Shelah, Saharon. A definable nonstandard model of the reals. J. Symbolic Logic 69 (2004), no. 1, 159–164.

The construction was slightly improved (in terms of weakening the hypotheses) in this 2018 publication in Journal of Symbolic Logic.


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