I'm working on a comparison between a set theoretical and an axiomatic construction of the hyperrational numbers $^*\mathbb Q$.

So far I have only found the construction of $^*\mathbb Q$ by using rational sequences on ultrafilters. But in the past some people explained to me that the hyperrationals can also be considered as a field extension by adjoining an "infinite" element $\omega$ to the field of rational numbers. One axiom, which is a property of that new element, would be $$\forall q \in \mathbb Q: \ \omega > q.$$

But this can't give us a full description of the field, i.e. there must be more axioms also telling us how the order relation $<$ is extended on $^*\mathbb Q$. In particular $^*\mathbb Q$ is uncountable while the field extension $\mathbb Q(\omega)$ is countable if we don't add additional axioms.

So is there any axiomatic system which fully describes the hyperrational numbers as a field extension of the rational numbers?

What I am aiming for is some kind of "check list" for the properties of the hyperrational numbers constructed by ultrafilters. That is, I want to show that the hyperrationals introduced by ultrafilters indeed satisfy all the necessary axioms in the same way as the real numbers introduced as equivalence classes of Cauchy sequences satisfy the axioms for a complete ordered field. Also, is there a term describing $^*\mathbb Q$ e.g. as "ordered non-standard field"?


Here is the recipe:

  • Take any complete first-order theory defining the rational numbers, in which every standard rational number is definable.
  • Add a new constant symbol "$\omega$"
  • For each rational number $q$, add an axiom "$\omega > q$"

The resulting theory is consistent, and thus has a model.

A simple recipe for the first point is:

  • Put whatever operators you want into the language (e.g. $+, \cdot, <$)
  • Add a constant symbol for every standard rational number $q$
  • For every true statement $P$ in this language about the standard model of the rationals, add $P$ as an axiom to this theory.
  • $\begingroup$ But how do you then get uncountability of *$\mathbb Q$ if you only add one constant for every element q in $\mathbb Q$? $\endgroup$ – user99680 Feb 14 '14 at 16:58
  • $\begingroup$ @user99680: you don't: I'm pretty sure there are countable nonstandard models too. To do non-standard analysis right this way, you really need to make a non-standard model of set theory, not just a non-standard model of the rationals or the reals. $\endgroup$ – Hurkyl Feb 14 '14 at 17:00
  • $\begingroup$ Is the last point you are mentioning the transfer principle (every true statement about $\mathbb Q$ also holds in $^*\mathbb Q$)? Also, wouldn't it be enough to only add the axioms for an ordered field as axioms to this theory, because all true statements about $\mathbb Q$ are derived from those axioms? (Sorry if that's a silly question.) Because then my check list would really just be the axioms for an ordered field, plus $\forall q\in\mathbb Q: \omega > q$. I was under the impression that the hyperrational numbers are unique (up to isomorphisms), not that there exist different models. $\endgroup$ – KDuck Feb 14 '14 at 17:21
  • $\begingroup$ I have seen "hyperreals" used both to describe a specific non-standard model of the reals, and as a general term for any suitable non-standard model of the reals. I assume "hyperrationals" would similarly be used both ways. I don't think I've ever seen someone try to give an (external) axiomatization of the specific non-standard model constructed by the ultrapower, although I haven't studied that perspective much. I imagine if such a thing existed, it would be something like "elementarily equivalent but not isomorphic to $\mathbb{Q}$, and has some suitable saturation properties". $\endgroup$ – Hurkyl Feb 14 '14 at 17:33
  • $\begingroup$ Thanks, that fully answers my questions! In that case I will just show that the non-standard model constructed by the ultrapower satisfies all axioms for an ordered field and additionally the existence of an infinite element. $\endgroup$ – KDuck Feb 14 '14 at 17:52

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