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 $\beth_1$?