# Tagged Questions

Model theory is the study of (classes of) mathematical structures (e.g. groups, fields, graphs, universes of set theory) using tools from mathematical logic. Objects of study in model theory are models for formal languages which are structures that give meaning to the sentences of these formal ...

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### How can be a set of partial isomorphisms defined from a n-back-and-forth system?

While studying partial Ebbinghaus-Flum's Mathematical Logic, I came across the partial isomorphism definition, as build upon an $n$-back-and-forth system. Consequently, the question I raise in the ...
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### Extended reals from ultraproduct of algebraic numbers

Let $\mathbb{A}$ denote the field of real algebraic numbers. Let $\mathcal U$ denote a free ultrafilter. Construct $F=\prod_{\mathcal U} \mathbb{A}$. This is a field containing $\mathbb A$, and we ...
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### Is there a sentence in the language $\{xRy\}$ with only infinite models?

Can you find a sentence in a language with only a binary relation $R$, all of whose models are infinite?
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### Question on existential sentences

A sentence is called existential if it is of the form $\exists x_1 \ldots \exists x_n \ \phi(x_1, \ldots, x_n)$, where $\phi$ is quantifier free. We know that (see Chang-Keisler "Model Theory", ...
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### define the reals in a non-archimedean elementary extension of the real field.

Can it be done? We have the real field $(\Bbb R,+,-,\times,0,1,<)$, of course $(0,1,-,<)$ are definable using the rest. We take an elementary non-archimedean extension. Can we define the ...
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### What's the best way to teach oneself both Category Theory & Model Theory?

I've done a bit of reading around both Category Theory & Model Theory (CT & MT) as a novice in each field. I'm interested in how they might combine, particularly when applied to Algebra. [So ...
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### Infinite set of standard primes as the set of standard prime divisors of a nonstandard number

Suppose $(N, +, \cdot, 0, 1, <, =)$ is a proper elementary substructure of $(N^*, +^*, \cdot^*, 0^*, 1^*, =^*, <^*)$. Show that there exists some (infinite) $b$, where $b ∈ N^*$, such that for ...
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### What is $R(\omega)$ (and where can I find definitions for similar common notation)?

Model Theory by Chang and Keisler references $R(\omega)$ frequently, usually in the context of models $\langle R(\omega), \in\rangle$ of ZF. What does this notation mean, specifically? From the ...
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### $\omega$-categoricity of a theory

Let $T$ be a complete $\omega$-categorical theory which have infinite models, and $C$ a $\omega$-saturated model of $T$. Let $A\subseteq C$ and $T_A$ be the theory of the model $C_A$, the structure ...
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### True and provably true sentences in a model. Are they the same thing?

In logic, it is said that each sentence in a (consistent) theory is either true or false in a given model. Checking the truth of a sentence in a finite model amounts essentially to finite enumeration ...