# Is the standard model of the reals the only model up to isomorphism of cardinality $\beth_1$? [duplicate]

TL;DR: I'm trying to figure out what you need to do to pin down the standard model of the reals without explicitly constructing such a model and then defining the standard model $$W$$ of the reals up to isomorphism as anything isomorphic to $$W$$ . I'm wondering whether cardinality alone is sufficient to pick out the standard model in this one specific case. I'm trying to keep the question focused because I don't understand the subject matter very well.

The Löwenheim-Skolem Theorem promises us, among other things, the existence of at least one model of each infinite cardinality $$\kappa$$ provided that a theory $$\sigma$$ has at least one model $$M_\lambda$$ of cardinality $$\lambda$$, where $$\lambda$$ is an infinite cardinal.

It promises more than just that, the models of various infinite cardinalities are related to each other in some way, but I don't really understand that part yet.

Let the signature $$\tau$$ refer to the axiomatization of the real numbers as an ordered field. I don't know whether the completeness axiom described in the link is expressible in a first order theory or not.

Note: I'm using an explicit construction of $$\mathbb{R}$$ here to satisfy the prerequisites of the L-S theorem, but only because I can't find an explicit representation of a countable model of $$\tau$$ and I don't know whether the computable reals would work here.

Let $$\mathbb{R}$$ be the standard model of the reals up to isomorphism, where each element in the domain is an equivalence class of Cauchy sequences of $$\mathbb{Q}$$ , where each $$\mathbb{Q}$$ is an equivalence class of pairs of an integer and a positive natural number. In this setting I think $$(+, -, *)$$ can simply be defined componentwise, multiplicative inverses can be defined using $$*$$, and $$<$$ can be defined by comparing the difference to the $$0_\mathbb{R}$$ . We also define what it means for a Cauchy sequence to be positive or negative, but only as a means of "implementing" comparison with $$0_\mathbb{R}$$ .

If we use $$\mathbb{R}$$ itself to demonstrate that $$\tau$$ has an infinite model, then we know there are models of every infinte cardinality.

$$\mathbb{R}$$ as described above has cardinality $$\beth_1$$. Is it the only thing with cardinality $$\beth_1$$?