# First-order Indistinguishibility of “the continuum”

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]

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Thanks!${}{}{}{}$ –  Amit Kumar Gupta Apr 4 '11 at 14:50