# Questions tagged [model-theory]

Model theory is the branch of mathematical logic which deals with the connection between a formal language and its interpretations, or models.

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### In finite model theory, is "invariant FOL with $\varepsilon$-operator" unavoidably second-order?

Throughout, all structures are finite. Say that a class of finite structures $\mathbb{K}$ is $\mathsf{FOL}_\varepsilon^\text{inv}$-elementary iff it is the class of finite models of a sentence in the ...
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### Natural Numbers

Let $L$ be a countable language and $M$ be a model of $L^N$ (the realization of $L$ in the natural numbers $N$) in which every recursive unary relation is expressible. Show that $M$ is not recursive.
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### Are some interesting mathematical statements minimal?

Gödel's set $\mathrm{L}$, of constructible sets, decides many interesting mathematical statements, as the Continuum hypothesis and the Axiom of Choice. Are some interesting mathematical questions, ...
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### The "first-order theory of the second-order theory of $\mathrm{ZFC}$"

$\newcommand\ZFC{\mathrm{ZFC}}\DeclareMathOperator\Con{Con}$It is often interesting to look at the theory of all first-order statements that are true in some second-order theory, giving us things like ...
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### Expressiveness in arithmetic

Let $\mathcal{S}$ be a formal system for arithmetic (e.g. $P$ or $PA$), $f:N^q\rightarrow N^p$ a function of $N^q$ on $N^ p$ and $\alpha(\mathbf{x})$ a formula of $\mathcal{S}$ with $p$ free variables....
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### Is $(\mathbb R_{\mathrm{an}}, (x\mapsto x^r)_{r\in\mathbb R})$ still the largest known polynomially bounded o-minimal structure so far?

From Chris Miller's paper in 1995, the structure $(\mathbb R_{\mathrm{an}}, (x\mapsto x^r)_{r\in\mathbb R})$,is the largest known polynomially bounded o-minimal structure as of that time. I wonder if ...
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### What is this quotient of the free product?

Previously asked at MSE. The construction here can generalize to arbitrary algebras (in the sense of universal algebra) in the same signature with the only needed tweak being the replacement of "...
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### Are the real numbers isomorphic to a nontrivial ultraproduct of fields?

Let $K_1, K_2, \dots$ be a countable sequence of fields, and let $\prod_{\mathcal F} K_i$ be the ultraproduct with respect to some nonprincipal ultrafilter $\mathcal F$. Question: Can there be a field ...
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### Finitely presentable groups are residually finite if and only if they are universally pseudofinite

Suppose $G$ is finitely presentable. Then residual finiteness of $G$ is equivalent to $G$ satisfying the universal theory of finite groups (equivalently, to every existential statement true in $G$ ...
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### Do maximal compact logics exist?

By "logic" I mean regular logic in the sense of abstract model theory (see e.g. the last section of Ebbinghaus/Flum/Thomas' book). My question is simple: Is there a logic $\mathcal{L}$ ...
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### Gödel coding and the function $z(x)$

The function $z(x)$ that associates to each formula $\alpha$ of $P$ its Gödel number $z(\alpha)$ is external to the system. How then can expressions in which $z(x)$ be involved be expressed in $P$? ...
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### Interpretation of model theory in algebraic geometry

I found a paper Some applications of a model theoretic fact to (semi-) algebraic geometry by Lou van den Dries. In this paper, the author uses model theoretical methods to prove the completeness of ...
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### Truth in a different universe of sets?

I understand that provability and truth as different concepts. Provability is syntactic, it only concerns whether the given sentence can be derived by reiterating the inference rules over a collection ...
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### Provability predicates

We know that there are provability predicates, that is, predicates derived from the recursive relation "x is a demonstration of y", with which Godel's second incompleteness theorem would not ...
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### Logical content of Gauss's Lemma (arithmetic)

In the context where $a$, $b$, $c$ are integers we have $(a \mid bc, a\land b = 1) \Rightarrow a\mid c$. This result is called Gauss's Lemma in French Highschool. It is well known that (Steve Awodey, ...
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### Classical first-order model theory via hyperdoctrines

I have been reading this discussion by John Baez and Michael Weiss and I find this approach to model theory using boolean hyper-doctrines very interesting. One of their goal was to arrive at a proof ...
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### Can this theory of dyadic rationals prove that multiplying by three is the same as summing thrice?

(This question arose from a discussion with Jade Vanadium about a theory of dyadic rationals.) Let $T$ be the 2-sorted FOL theory with sorts $ℕ,ℚ$ and constant-symbols $0,1$ and binary function-...
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### Seeking clarification of ultrapower nonstandard model of arithmetic

I've read that one nonstandard model of arithmetic is: take $\mathbb{N}^\mathbb{N}$, the set of countably infinite sequences of natural numbers take a quotent that gives the ultrapower: identify ...
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### If a theory has many mutually non-embeddable countable models can it have a countable $\omega$-saturated model?

A theory can have $2^\omega$-many non-isomorphic countable models but has a countable $\omega$-saturated model. (https://math.stackexchange.com/questions/305602/if-a-theory-has-a-countable-omega-...
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