Continuous First order logic vs Second order logic

Question

By continuous first order logic I mean first order logic but replace truth values taking on 0 or 1 with the compact set [0,1].

Is continuous first order logic strong enough to make statements about all subsets? Example: every bounded non-empty subset has a supremum.

Is there some trick where this can be done?

https://arxiv.org/pdf/0801.4303.pdf

Answer 1

Continuous logic (as described in http://math.univ-lyon1.fr/~begnac/articles/mtfms.pdf, for example) does not allow you to make statements about arbitrary subsets. There are (at least) two ways that this is true:

1. The quantifiers in continuous logic range over elements of the structure, not over subsets. Thus there is no direct way that you can say something like $\forall S \subseteq M \ldots$ at all.
2. Given any formula $\varphi(x_1, \ldots, x_n)$ and a structure $\mathcal{M}$ of the appropriate language, the set $\{(a_1, \ldots, a_n) \in M^n : \mathcal{M} \models \varphi(a_1, \ldots, a_n)\}$ is a closed subset of $M^n$. In this sense the language of continuous logic cannot describe any non-closed set (it can't necessarily describe all closed sets either, nor even properly describe all zerosets, but that's a more subtle matter).

You mentioned order-completeness in the question. Often you can't even put an ordering into the language (you would have to put the distance to the ordering as a predicate instead, and then uniform continuity requirements will impose some compatibility between the ordering and the metric).

On the other hand, metric completeness (every Cauchy sequence converges) is definitely not expressible in continuous logic. Indeed, if you have any metric structure $\mathcal{M}$ where the underlying metric space $(M, d)$ is incomplete*, and let $\overline{\mathcal{M}}$ be the (unique, up to isomorphism) extension of $\mathcal{M}$ based on the metric completion of $(M, d)$, then $\mathcal{M} \preceq \overline{\mathcal{M}}$.

*: Usually the term "metric structure" is reserved for structures based on complete metric spaces, so I'm abusing the name a little bit here. I hope the meaning is clear enough.