# 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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### Morley’s Categoricity Theorem for uncountable languages.

Where can I find an accessible exposition of Shelah’s generalization of Morley’s theorem to uncountable languages? (Please, do not answer “Shelah’s Classification Theory”.)
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### Why are structures with no relations called algebras?

"If [a given structure] A has no relations it is termed an algebraic structure, or simply an algebra" - Wolfgang Rautenberg, A Concise Introduction to Mathematical Logic, 3rd edition, page 42. I ...
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### Show For any language L two L-structures M and N are elementarily equivalent iff they are elementarily equivalent for every finite sublanguage.

Setting For any language $\mathcal L$, two $\mathcal L$-structures $\mathcal M$ and $\mathcal N$ are elementarily equivalent iff they are elementarily equivalent for every finite sublanguage. ...
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### Comparing Category Theory and Model Theory for Master's Thesis.

I am currently doing a Masters thesis in pure maths, and the two current fields that excite me are Category Theory (CT) and Model Theory (MT). I have been reading up on David Marker's Model Theory: ...
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### Are the implicitly definable sets of a second-order theory the sets the second-order quantifiers range over?

I know that in a second-order setting, due to the failure of the Beth definability theorem, implicit and explicit definition come apart (i.e., there are predicates which can be implicitly, but not ...
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### Given L = {<,c0,c1,…} and T3 the theory of DLO with sentence asserting co < c1 < …, Show T4 is complete with four countable models.

Let $\mathcal L_3 = \{<,c_o,c_1,\ldots\}$, and $T_3$ be the theory of DLO with sentences added stating $c_o < c_1 < \ldots$. Now let $\mathcal L_4 = \mathcal L_3 \cup \{P\}$, where $P$ is a ...
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### Link between definitional expansions and definitional extensions.

I need to prove this, Let $T$ be a theory in language $L$, let $T'$ be a definitional extension of $T$ to language $L\subseteq L'$. If $\mathcal {M} \models T'$, then $\mathcal M$ is a ...
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### What are the L-sentences that are true in an empty structure?

I am looking for an algorithm or set of rules to figure out whether a sentence (in first order logic) is true when we are dealing with an empty set as domain. Clearly, it has to be a sentence (no free ...
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### (stability-theoretic) ¨weakly normal groups" are closed under subgroups

Let me first introduce two definitions: For a structure $\mathcal{M}$ in a language $\mathscr{L}$ and a subset $X \subseteq M^n$, the fully induced structure on $X$ is a structure $\mathcal{X}$ with ...
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### Why do we need ultrafilters to make sense of the cartesian product of $\mathcal{L}$-structures

I'm trying understand why we need ultrafilters in model theory. Here is how I see things. Could someone tell me if this is correct ? Further explanations are always welcome. Let $\mathcal{L}$ be a ...
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### Multiplicative reducts of fields an elementary class?

Consider the multiplicative reducts of fields, that is fields except the addition operation is removed. We are considering the signature {*}, where * is an operator of arity 2. Is that class an ...
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### Spectrum restrictions in the signature consisting of just a single binary operation

In the signature {*}, where * is an operator of arity 2, is there a theory whose spectrum is the set of prime powers?
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### embdedding standard models of PA into nonstandard models

Maybe it's well known to experts, but is there always an embedding $f$ of the standard model of Peano arithmetic into a nonstandard model? By Peano arithmetic I mean its first-order version, with the ...
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### prove Lindenbaum’s lemma for a countable language

Been reading through some model theory and got to a section on constructing models from syntax and i have been presented with the following problem, sorry for the lack of solution i just have no idea ...
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### What is the name of the set models can be drawn from?

What is the name of the set models can be drawn from? For example in propositional calculus an assignment function $v : P \rightarrow \{T,V\}$ can be the model of a formula $a$. What is the (generic) ...
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### Number of Ways of Combing Linear Orders

I have a slight variant of this question. I would also appreciate any references for questions like this. (The question is inspired by the study of linear orders in model theory.) Suppose you're ...
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### Models of the empty theory T, and proof that T $\kappa$ categorical for every cardinality. [duplicate]

Bombarding stack exchange with model questions today I am tackled with the following problem: Note this is the same question as posted by B0bg0blin's here, i just need a bit more clarity. In the ...
The questions in model theory I am trying to tackle is: Show that there is a countable model of $Th(\langle \mathbb{R};+,.,-,-,1,< \rangle)$ which is non archimedean. Honestly i dont really know ...