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This is a followup to this question. Consider $\mathbb{C}$ as a structure - in the sense of first-order logic - with the graphs of addition and multiplication. Let $\mathcal{X}$ be the substructure wi …

I don't know a reference, but here's a (slightly overkill) proof: It's known that $(\mathbb{Z};+,\times)$ interprets an algebraically closed field $K$ of infinite transcendence degree and characterist …

This question is very close to this old MSE question of mine, which is still unanswered. Is there an (ideally reasonably-natural!) expansion of the structure $(\mathbb{N};+)$ in a finite language whos …

Below, all structures are finite, in a finite language, with underlying set an initial segment of the natural numbers. This has been edited to fix errors pointed out by Emil Jerabek in his answer belo …

(See e.g. here for background on the Eudoxus reals, which motivates this question.) Let $\mathcal{Z}=(\mathbb{Z};+,<)$. Say that a Eudoxus function is an $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that …

The answer is yes. I believe the following is best attributed to Feferman 1960; more generally, look up "arithmetized completeness theorem" (which is annoyingly different from the arithmetic completen …

For a computable$^*$ ordinal $\alpha$ (viewed as an $\{<\}$-structure), say that a first-order sentence $\sigma$ in a relational language $\Sigma\supseteq \{<\}$ decidably clarifies $\alpha$ iff $\alp …

Suppose $\mathscr{V}$ is a pseudovariety in a countable language $\Sigma$. Say that a minimal description of $\mathscr{V}$ (in the sense of Eilenberg/Schutzenberger rather than Reiterman) is a pair $( …

See Eilenberg/Schutzenberger, On pseudovarieties for background on pseudovarieties. I've phrased things in terms of pairs-of-sets to avoid some annoying language about multisets. Also, I'm aware that …

This is motivationally related to an earlier question of mine. Given a first-order theory $T$, let $\widehat{D}(T)$ be the semiring defined as follows: Elements of $\widehat{D}(T)$ are equivalence c …

This question came up while reading the paper Hales, What is motivic measure?. Broadly speaking, I'm interested in which ideas from motivic measure make sense in arbitrary first-order theories (or eve …

Given a clone $\mathcal{C}$ over $\{\top,\perp\}$, let $\mathsf{FOL}^\mathcal{C}$ be the version of first-order logic with connectives from $\mathcal{C}$ in place of the usual Booleans. Given a clone …

Given a clone $\mathcal{C}$ over the set $\{\top,\perp\}$, let $\mathsf{FOL}^\mathcal{C}$ be the version of first-order logic with (symbols corresponding to) elements of $\mathcal{C}$ replacing the us …

The following question arose while trying to read Shelah's papers Models with second-order properties I-V. For simplicity, I'm assuming a much stronger theory here than Shelah: throughout, we work in …

It's well-known$^*$ that the Los-Tarski theorem ("Every substructure-preserved sentence is equivalent to a $\forall^*$-sentence") fails for $\mathsf{FOL}$ in the finite setting: we can find a first-or …