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The construction I've seen of the field of hyperreal numbers considers a non-principal ultrafilter $\mathcal{U}$ on $\mathbb{N}$, then takes the quotient of $\mathbb{R}^{\mathbb{N}}$ by equivalence respect to $\mathcal{U}$, that is, $(a_{n})=(b_{n})$ if the set $S$ of indices $n$ for which $a_{n}=b_{n}$ is inside $\mathcal{U}$.

Is it clear (or even true) that one obtains an isomorphic field with a different choice of ultrafilter?

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  • $\begingroup$ No, it's not clear, and I think that this is either open or blatantly false. I'm not sure, though. $\endgroup$ – Asaf Karagila Mar 20 '14 at 0:54
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    $\begingroup$ One can get a wide variety of non-isomorphic ultrapowers even if the index set is fixed. There is a large literature. Useful, because we can get additional "nice" properties by appropriate choice of ultrafilter. $\endgroup$ – André Nicolas Mar 20 '14 at 1:00
  • $\begingroup$ related: mathoverflow.net/questions/88292/… $\endgroup$ – Ben Crowell Mar 20 '14 at 1:30
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    $\begingroup$ If the continuum hypothesis holds, then yes. You can read more about this in the following link: mathoverflow.net/questions/136720/… $\endgroup$ – user136666 Mar 20 '14 at 1:37
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No, one has to invoke rather strong hypotheses such as the Continuum Hypothesis (CH) to obtain uniqueness. The classic paper on this is the following, where it is proved that it is consistent with ZFC that there are $\,2^{\large \aleph_0}$ non-isomorphic hyperreal fields.

Judy Roitman. Non-Isomorphic Hyper-Real Fields from Non-Isomorphic Ultrapowers.
Math. Z. 181, 93-96 (1982)

See also N. Aldenhoven's 2010 Bachelor's thesis Uniqueness of the Hyperreal Field where you can find an elementary exposition of an earlier result of Erdos, Gillman and Hendrikson (1955) that all real-closed fields of the same cardinality as $\,\Bbb R\,$ with a $\eta_1\!$-ordering are isomorphic.

More recent results can be located by searching for citations of Roitman's paper.

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  • $\begingroup$ I wouldn't call CH a "strong hypothesis". (The term "strong" has a consistency strength ring to it, whereas the consistency of CH and its failure are both equivalent to the consistency of ZFC to begin with.) $\endgroup$ – Asaf Karagila Mar 20 '14 at 1:29
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    $\begingroup$ @Asaf, I think Bill is using "strong" in the sense of "controversial" rather than in the foundational sense. $\endgroup$ – Mikhail Katz Apr 29 '14 at 8:45
  • $\begingroup$ The second link is broken now. $\endgroup$ – KCd Apr 9 '17 at 21:55
  • $\begingroup$ @KCd Updated, thanks. $\endgroup$ – Bill Dubuque Apr 10 '17 at 0:27
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I would like to add to Bill's answer:

Judy Roitman found a particular model of ZFC$+ \neg$CH where there are $\mathfrak c$ many different hyperreal fields. Very recently, it was shown that in every model of ZFC$+ \neg$CH there actually exist $2^\mathfrak c$ different hyperreal fields... as many as you could hope for since there are exactly $2^\mathfrak c$ (free) ultrafilters on $\omega$!

A dichotomy for the number of ultrapowers

Ilijas Farah & Saharon Shelah

Journal of Mathematical Logic 10 (01n02):45-81 (2010)

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    $\begingroup$ Very recently? [citation needed]. $\endgroup$ – Asaf Karagila Jun 17 '14 at 1:45
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To obtain the uniqueness of the hyperreal field one needs to add the axiom of saturation which is arguably similar to completeness (in the case of the real field). Furthermore, like the real (complete Archimedean) field, the hyperreals have a definable model. For a discussion see Keisler's text on foundations here.

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