First-order Indistinguishibility of "the continuum"

Question

Let us consider two different models of the continuum $\mathbb{R}$ (that is, we take two arbitrary ZF-models, and we look at the continuum in each one of these models).

Let us now suppose that we have these two models enriched with additional structure (for example with addition, product, topology, etc.). Under which conditions of this additional structure is it known that these two models (over $\mathbb{R}$) are elementary equivalent (i.e., they satisfy the same first-order sentences in the enriched language)?

Let me consider an example to explain better my question. If we look at these two models with just addition and product we know that they are elementary equivalent because both models are real-closed fields (which is a complete first-order theory).

Is there some general statement that tells us that using first-order sentences we will not be able to distinguish these two models (even with some enriched vocabularies)? For example, what happens if we enrich the models with the standard topology? [Here I am thinking on a two-sort first-order language where one sort is for real numbers, and the other for the topological opens]

Answer 1

If you add a multi-sorted structure that allows you to talk about measure, and have a unary predicate which applies to sets of reals which says whether a set is measurable or not, you can have non-equivalence because in some models of ZF, all sets of reals are measurable, and in others they aren't.

Also, you could have one of your models be such that its naturals are non-standard, where moreover in one of the models, Con(ZF) is a true arithmetical sentence, and in the other its false. Then in each model, I think you can have the reals interpret the natural numbers and their arithmetic, and these structures would also be elementarily inequivalent.

I think there's a few ways to interpret your question and make it precise. One of the things it depends on, I think, is what you mean by "the reals."