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I understand that reals are defined as "completing" the rationals, which (at least in ZFC) are in turn derived from the natural numbers.

So, if ordinal numbers are viewed as an extension of the natural numbers, are there constructs of ordinals corresponding in the same way as rationals and reals are derived from natural numbers ?

If there are, then does this have any serious implications for models of reality which seem to be predominantly based on the reals ?

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  • $\begingroup$ Since when mathematicians care about models of reality?? (Also, if you really consider the expansion of the universe, it seems more reasonable to use an ultrametric space for this modeling, e.g. the $p$-adic numbers.) $\endgroup$
    – Asaf Karagila ♦
    Commented Mar 30, 2014 at 22:20
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    $\begingroup$ Well, if you add infinite elements in a certain way (different from ordinals) and take quotients and complete, you get $\mathbb R[\epsilon]$, the starting point for nonstandard analysis. $\endgroup$
    – Alex Becker
    Commented Mar 30, 2014 at 22:21

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These are called Surreal Numbers. It is a proper-class sized order field which is complete, in the sense that every subset has a least upper bound and greatest lower bound.

It embeds all the ordered fields in the universe [of set theory], and all the ordinals too. It does not, however, extend ordinal arithmetic since ordinal addition is not commutative.

There are also non-standard fields which extend the real numbers, like the hyperreal numbers, and more. But these are "mere sets". So they do not extend the ordinals.

Finally, let me point out that mathematicians, in particular set theorists and model theorists don't consider their mathematics as a model of the physical universe. We keep the church and state separate from our end.

Moreover, if you consider the expansion of the universe and how every point look outwards thinks that it is in the center of the universe, wouldn't it be much more fitting to use some ultrametric space to model the universe (i.e. a space where every point inside an ball is its center) rather than Euclidean metric spaces? (For example the $p$-adic numbers seem like a good candidate for that.)

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  • $\begingroup$ The limit ultrapower construction gives a class-size hyperreal field which is isomorphic to the maximal surreal field (in NBG with global choice), and in addition you gain the transfer principle. So it is incorrect to assert that the hyperreals are "mere sets". $\endgroup$
    – Mikhail Katz
    Commented Apr 3, 2014 at 14:27
  • $\begingroup$ @user72694: If you switch set theory, you can change a lot of propositions. For example it's obvious that $x\in x$ is true; oh wait, I didn't tell you that I'm working in $\sf ZF-Fnd+AFA$. Yes, in $\sf NBG$ you can do more. In $\sf MK$ you can do even more. If you continue to extend your abilities to form collections you can do even more. As for me, I'm still working with a single sort theory of sets, more specifically $\sf ZFC$ (and extensions thereof) unless stated otherwise. $\endgroup$
    – Asaf Karagila ♦
    Commented Apr 3, 2014 at 14:41
  • $\begingroup$ But the standard presentation of the surreals is precisely in NBG, so I am not sure what you are arguing. Whichever framework you can develop a class-size surreal field, you can also develop a class-size hyperreal field. To prove that the surreal field is maximal, you need NBG. But once you commit yourself to NBG, the hyperreals also become maximal. What you gain is the transfer principle which makes it a useful tool in analysis and other applications. $\endgroup$
    – Mikhail Katz
    Commented Apr 3, 2014 at 14:48
  • $\begingroup$ ... and of course the "proper-class sized order field" is not formalizable in ZFC, the single sort theory you are still working with. $\endgroup$
    – Mikhail Katz
    Commented Apr 3, 2014 at 18:12
  • $\begingroup$ And of course that it is; as a schema of statements that we can easily identify from the meta-theory. $\endgroup$
    – Asaf Karagila ♦
    Commented Apr 3, 2014 at 18:30
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The complex numbers are the completion of the reals in the algebraic sense, by which I mean the set of solutions to polynomials in $\mathbb{R}[x]$ is $\mathbb{C}$.

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  • $\begingroup$ thanks. While this is true, it is the direction that Asaf gives that I was interested in. $\endgroup$
    – Tom Collinge
    Commented Mar 31, 2014 at 8:26

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