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One useful consequence of $ZFC$ is that the real numbers can be shown to be unique upto isomorphism.

According to wikipedia:

The use of the definite article the in the phrase the hyperreal numbers is somewhat misleading in that there is not a unique ordered field that is referred to in most treatments. However, a 2003 paper by Vladimir Kanovei and Shelah shows that there is a definable, countably saturated (meaning ω-saturated, but not of course countable) elementary extension of the reals, which therefore has a good claim to the title of the hyperreal numbers. Furthermore, the field obtained by the ultrapower construction from the space of all real sequences, is unique up to isomorphism if one assumes the continuum hypothesis.

Unfortunately the page does not give a reference for this result. The result is referenced though in this question. But assuming it correct, presumably this not a good enough reason to settle the continuum hypothesis. But I find the generalisation of uniqueness of the reals upto isomorphism to the hyper-reals here interesting and suggestive.

Q. Why would one want to discount this being of significance in settling the continuum hypothesis in the positive?

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Mozibur Ullah
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    Because forcing axioms seem more important than the hyperreal numbers; at least from a set theoretical point of view. Your argument would be equal to adopting $V=L$ just for the sake of settling Whitehead's problem positively. – Asaf Karagila Feb 17 '14 at 08:14
  • Ok, fair enough - it probably means I don't know enough set theory:). When you say forcing axioms do you mean by that the large cardinals? – Mozibur Ullah Feb 17 '14 at 08:32
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    No. Forcing axioms (while related to large cardinals sometimes) are axioms stating that given certain partial orders [read: forcing notions] which have particular properties, then given a collection of "some" dense subsets of such forcing notion, there is a filter which meets all of this collection. I'm being very vague here. The point is that one of the most natural axioms in this context already implies $2^{\aleph_0}=\aleph_2$. And I would feel that from a set theoretical point of view, forcing axioms are more important than hyperreal numbers. – Asaf Karagila Feb 17 '14 at 08:34
  • Ok. So from a set theoretic perspective forcing notions are more natural than large cardinal axioms. Which 'natural axiom' settles $2^{\aleph_0}=\aleph_2$? – Mozibur Ullah Feb 17 '14 at 08:49
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    No, it's not "more natural". Large cardinals are very natural and important set theoretically, and so is forcing. It's like saying that oxygen is more important and natural than ATP to our human survival. Both are important and needed. Large cardinals, however, rarely interact with things as small as the continuum. On the other hand, forcing axioms do. $\sf PFA$ (Proper Forcing Axiom) settles the continuum to be $\aleph_2$, as do its strengthening $\sf MM$ (Martin's Maximum), and several other related axioms of similar flavor. – Asaf Karagila Feb 17 '14 at 08:53
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    See also this somewhat related question. – user642796 Feb 17 '14 at 09:15

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