# 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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### Limits of finite structures - first order logic

Assume that $\mathcal{C}=\{M_i:i\in I\}$ is an infinite collection of different finite $\mathcal{L}$-structures in a first-order language $\mathcal{L}$. The question is: What kind of infinite "...
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### Lemma (?) on syntactically saturated maximal sets of sentences and existential sentences

Is there a proof of the following claim? $$\not \vdash_{{\rm FS}(L,M)} \exists x ~\alpha \implies [\alpha][x/c] \tag{T}$$ where no variable other than $x$ occurs free, $c$ is a name, ${\rm FS}$ ...
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### What is the importance of “variety of algebras” in Universal Algebra?

Given an algebraic category, Birkhoff's Variety Theorem gives a categorical characterization of the full subcategories whose object-class forms a variety (i.e. can be defined by equations in the sense ...
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### Complete theory with a infinite model has only infinite models

Let $T$ be a complete first order theory with a infinite model $M$. I want to show that every model $N$ of $T$ must be infinite. Since $T$ is complete then for every sentence $\phi$ we can either ...
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### Algebraic numbers in real closed fields

I have been looking at the discussion of real closed fields in Appendix B of Marker's Model Theory:an Introduction. I am baffled by what it says about the uniqueness of real closures. I have no ...
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### I don't really understand what a model is.

I've studied a bit of first order logic, and I still don't understand what a model really is. A model of a theory $T$ is an interpretation which assigns the value True to its sentences. Ok, that'...
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### Can type theory be viewed as an alternative to model theory?

While type theory certainly has traditionally been used for different purposes than model theory, as noted in this Philosophy SE post, I wonder to what extent type theory could model model theory ...
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### Structure/Model for first order language

For simplicity lets say $L$ is a first order language with a single function symbol $f$ and no other nonlogical symbols What would a structure/model, $M$ say, for $L$ look like? I know $M$ has some ...
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### If $\cal{I}$ is a indiscernible sequence over $A$ then it is indiscernible over $acl(A)$

Let $\cal{I}=(b_i\mid i\in I)$ be an infinite indiscernible sequence over $A$. And let $acl(A)$ be the algebraic closure of $A$. (all in some structure) I am trying to show that $\cal{I}$ is also ...