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Somebody told me it's possible to show the existence of rings non-isomorphic to any subring of the real numbers using model theory, but they couldn't get an example.

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Via model theory this is trivial: the theory of ordered rings is first-order, so in particular via Löwenheim–Skolem it has models of every infinite cardinality.

But this is much more elementary. Simply take your favourite ordered ring $R$, and consider the polynomial ring $R[X]$, and order it lexicographically${}^\dagger$. Then $R[X]$ is an ordered ring which is non-Archimedean: for every $n$ you have $n\lvert X\rvert<1$, but $X\neq 0$. This is not possible in a subring of real numbers (not with the standard ordering).

In fact, the same construction can be used to construct ordered rings of arbitrary (infinite) cardinality $\kappa$: you can just extend the order on $R$ to an order on the ring of polynomials in $\kappa$ variables, and for $\kappa> 2^{\aleph_0}$, this ring will not even be algebraically isomorphic to a subring of real numbers, simply because there are not enough reals.

In a way, this is an overkill: you can show that already ${\bf R}[X]$ is not algebraically isomorphic to a subring of the real numbers, even though it has the right cardinality. This is because ${\bf R}$ knows its own ordering: the positive elements are exactly the squares. In particular, the only ring homomorphism ${\bf R}\to {\bf R}$ must be constant, and thus onto (it must be constant on rationals for trivial reasons, and for any other real, the image is determined by the image of rationals below it). It follows that any ring homomorphism ${\bf R}[X]\to {\bf R}$ is not injective.


$\dagger$: As suggested by user49640 in the comments, when $R$ is a field, the ordering in $R[X]$ can be described another way: we can identify a polynomial $p\in R[X]$ with the polynomial function and say that $p<q$ if $p(r)<q(r)$ asymptotically as $r\to 0^+$.

Note also that we could have very well looked at asymptotic behaviour as $r\to +\infty$, and that would yield a different ordering of $R[X]$, which is also non-Archimedean, since then for any $n$ we would have $n\cdot 1<\lvert X\rvert$. This corresponds to the lexicographic ordering with the coefficients written starting from the higher orders, as suggested by KCd in the comments.

The two orderings of $R[X]$ can also be intuitively described as simply considering $X$ to be an infinitesimal or (respectively) infinitely large element. That way, it is clear that it is non-Archimedean.

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  • $\begingroup$ Clarify what you mean by lexicographic ordering: in what order are you writing terms in a polynomial to decide what is bigger? When I first read "lexicographic ordering" I thought $nX > 1$ for all positive integers $n$. $\endgroup$ – KCd Apr 6 '17 at 3:56
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    $\begingroup$ One interpretation of the ordering on $\mathbf{R}[X]$ is to declare $f(X)$ to be positive if $f(x) > 0$ when $x \to 0^{+}$. $\endgroup$ – user49640 Apr 6 '17 at 5:36
  • $\begingroup$ @KCd: I meant that you write a polynomial $a_0+a_1x+a_2x^2+a_3x^3+\ldots+a_nx^n$ as a sequence $(a_0,a_1,\ldots)$ and then order accordingly. In a way, that the other ordering is maybe more natural since usually we write the highest power first, but it has the disadvantage that you consider strings starting arbitrarily far in the past rather than ending arbitrarily late in a future -- I guess that is an aesthetic preference. Note though that it really does not matter, you get a non-archimedean ring either way, and you can get one from the other by considering $R[X^{-1}]\subseteq R(X)$. $\endgroup$ – tomasz Apr 6 '17 at 11:12
  • $\begingroup$ @user49640: I'm not exactly sure what you mean, but there is a similar description of the ordering. I will add it to the answer, thanks for the feedback! $\endgroup$ – tomasz Apr 6 '17 at 11:14
  • $\begingroup$ I like to think of this ordering as behavior when $X\rightarrow 0^+$, as @user49640 says. For me the more natural "first" ordering on $\mathbf R[X]$ is growth at infinity where $X > n$ for all integers (or even real numbers) $n$. All orderings on the field $\mathbf R(X)$ are obtained by looking at growth as $X$ tends to a real number from the right or from the left, or as $X$ tends to $\pm\infty$. $\endgroup$ – KCd Apr 6 '17 at 13:38
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Another example (of a non-Archimedian ordered field) worth checking out is David Hilbert's example in Section 12 of Foundations of geometry.

It is, to paraphrase, what you get by taking $\mathbb Q$, a single variable $t$, and then generating a field that is closed under the additional operation $x\mapsto \sqrt{1+x^2}$.

Viewed as real-valued functions, it can be proven that each such expression is eventually positive or eventually negative, and then the ordering is that $f\leq g$ if there exists an $b\in \mathbb R$ such that $f(x)\leq g(x)$ for all $x\in [b,\infty)$.

Hilbert's original description is a little archaic to a modern reader, but it's not too bad, and I'm sure there are more modern versions of it lying around somewhere.

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An example of non-archimedean field via ultraproducts. The cartesian product of fields $${\Bbb R}^{\Bbb N} = \Bbb R\times\Bbb R\times\Bbb R\times\cdots$$ isn't a field (isn't a model of...) because has zero divisors: $$(0,1,0,1,\dots)(1,0,1,0,\dots)=(0,0,0,0,\dots).$$ The solution is taking a quotient: let be $\mathcal U$ a nonprincipal ultrafilter on $\Bbb N$. Define $$(a_1,a_2,\dots)\sim(b_1,b_2,\dots)$$ when $$\{n\in\Bbb N\,|\,a_n=b_n\}\in\mathcal U$$ The quotient ${}^*\Bbb R = {\Bbb R}^{\Bbb N}/{\sim}$ will be a field with infinitesimals, like the class of equivalence of the sequence $$(1,1/2,1/3,1/4,\dots).$$

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  • $\begingroup$ Obtaining a non-archimedean field from a non-archimedean ring is easy, because you can extend the ordering to the field of fractions in the obvious way: namely $\frac{p_1}{q_1}\leq \frac{p_2}{q_2}$ iff $p_1q_2\leq p_2q_1$ (for all numerators and denominators positive). $\endgroup$ – tomasz Apr 5 '17 at 18:07
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    $\begingroup$ Usually the asterisk comes before the $\Bbb R$. (I've also put the \sim command in braces to fix the spacing.) $\endgroup$ – Asaf Karagila Apr 5 '17 at 18:34

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