# Some questions about the order on the hyperreals and the Ramsey theory on it.

Let $$^* \mathbb R$$ denote the set of hyperreal numbers, which is constructed with a nonprincipal ultrafilter on $$\mathbb N$$. Since we can order-embed $$\omega_1$$ into $$^* \mathbb R$$, by the Erdős-Dushnik-Miller theorem, we have $$^* \mathbb R \rightarrow (\omega_1, \omega)^2$$. I have $$3$$ questions regarding the order on hyperreal numbers and the Ramsey theory on it:

1. What order-types can we realize in $$^*\mathbb R$$? (I know that this question is not very precise but, for example, $$\omega + (\omega^* + \omega) \cdot \eta$$ is realized in any nonstandard model of arithmetic, where $$\omega^*$$ denotes the converse of $$\omega$$, and $$\eta$$ is the order-type of the rational numbers with the usual ordering, so I am expecting an answer like this but for $$^*\mathbb R$$).

2. Can we order-embed any ordinal $$\alpha < \omega_2$$ into $$^* \mathbb R$$? Note that since any hyperreal interval $$(a,b)$$ with real endpoints is order-isomorphic to $$^* \mathbb R$$, we can extract countably many $$\omega_1$$’s from $$^*\mathbb R$$, but can we do better than that?

3. Finally, for what ordinals $$\alpha < \omega_2$$ do we have $$^* \mathbb R \rightarrow (\alpha, \omega)^2$$? For the simplest case, do we have $$^* \mathbb R \rightarrow (\omega_1 + 1, \omega)^2$$? Or even $$^* \mathbb R \rightarrow (\omega_1+1, 3)^2$$? (Note that $$\omega_1 + 1$$ order-embeds into $$^* \mathbb R$$ by (2).)

It can also be interesting to see if we can further generalize (3) by increasing the second index from $$\omega$$ to some bigger countable ordinal $$\alpha$$ (note that it cannot be uncountable since $$2^{\aleph_0} \nrightarrow (\aleph_1, \aleph_1)^2)$$, thank you.

Since $$^*\mathbb{R}$$ is $$\aleph_1$$-saturated, every linear order $$L$$ of cardinality at most $$\aleph_1$$ embeds in $$^*\mathbb{R}$$. In particular, every ordinal $$<\omega_2$$ embeds in $$^*\mathbb{R}$$.

Proof: Enumerate $$L$$ by $$\omega_1$$. Now build the embedding one element at a time by transfinite induction. At each successor step, we only have to fill cuts in countable subsets of $$^*\mathbb{R}$$, which we can always do.

• Can we answer the first question with a “nice” form of an order-type?
– Tan
Jan 22, 2022 at 14:04
• @Tan I'm not sure what you mean. Name any order type you like of cardinality at most $\aleph_1$. It embeds. Jan 22, 2022 at 14:05
• @Tan Note that the same result (any linear order of size $\aleph_1$ embeds) holds for ultrapowers of the natural numbers. The weaker result about $\omega+(\omega*+\omega)\cdot \eta$ holds for arbitrary nonstandard models of arithmetic (which might be countable). Similarly, since the hyperreals are an ultrapower, we get $\aleph_1$-saturation. This is not true for arbitrary nonstandard models of the theory of the real numbers. Jan 22, 2022 at 14:23
• @Tan Certainly. There are countable models of this theory, like the real algebraic numbers. But maybe by "nonstandard model", you mean a proper elementary extension of the reals. Then consider, for example, the field of Puiseux series over the real numbers. This field still has a countable subset which is dense in the order topology (the Puiseux monomials with coefficients in the rationals), so it fails to embed $\omega_1$. Jan 22, 2022 at 15:28