# 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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### Explicit countable elementary extension of $\mathbb{N}$

I would like to see an explicit example of a non-trivial elementary extension of the structure $(\mathbb{N}, +, \cdot, 0, 1)$ where $\mathbb{N}$ includes zero. Most of all I am interested in countable ...
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### How does Gödel's second incompleteness apply to any theory containing arithmetic?

If I understand correctly, there are two facts proven by Gödel's second incompleteness theorem, for a formal theory containing arithmetic 1) It is possible to express the consistency of the theory ...
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### Quantifier elimination for theory of equivalence relations

Let $\mathcal{L}=\{\sim\}$ and $\Sigma_\infty$ be the set of axioms stating that: (i) $\sim$ is an equivalence relation (ii) Every equivalence class is infinite (iii) there are infinitely many ...
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### Regarding the theory $REI_{\alpha}$

The theory $REI_{\alpha}$ has as its language $L=\{E_{\beta}|\beta\leq\alpha\}\cup{\{E_{-1}\}}$, and each $E_{\beta}$ and $E_{-1}$ are binary relation symbols. Let $T$ (=$REI_{\alpha}$) be the theory ...
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### Prove that formule is true in every realisation of language [duplicate]

I have language $\mathcal L$= $\{$P,Q,R$\}$, arity of $P,Q,R$ is $1,1,2$. And I have two formulas: 1) ($\forall x$ $P(x)$ $\land$ $\forall x$ $Q(x)$) $\leftrightarrow$ $\forall x$ ($P(x)$ $\land$ ...
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### Positive existential theory of an extension of the ring

When we know that the positive existential theory of a ring $R[x]$ in a language $L$ is undecidable, does it follow that the positive existential theory of $R[x,y]$ in the same language $L$ is also ...
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### Parameters and strongly minimal sets

Suppose $T$ is a countable complete theory, with monster model $\mathbb{C}$. A definable set $D := \phi(\mathbb{C}, \overline a)$ is strongly minimal if given any other formula $\psi(x, \overline b)$ ...
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### How to show $T=\{\forall x(x\neq S^{n}x)|1\leq n\}\cup\{\sigma\}$ in $\mathcal{L}=\{S\}$ is $\kappa$-categorical

Problem saying: Let $T=\{\forall x(x\neq S^{n}x)|1\leq n\}\cup\{\sigma\}$ in $\mathcal{L}=\{S\}$ , where $S$ is a unary function and $S^{n}$ abbreviates $\underbrace{S\dots S}_{n}$ , and ...
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### Can there be a countable transitive model satisfying the same $MK$ theory as $V$?

A little while ago, I asked whether or not there could be a countable transitive model satisfying the same $ZFC$ theory as $V$ (assuming that we're working within some $V$, or (if you like) that there ...
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### Defining a formula using FO+TC

Define a signature Σ and an FO+TC formula ϕ over Σ, such that: there is no infinite structure satisfying ϕ for every even natural number n>0 there is a structure of size n satisfying ϕ for every ...
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### model theory & algebraically closed field

recently I'm taking mathematical logic course, and the class covers some basic model theoretical ideas. Since I have not taken any abstract algebra course, It is so hard to understand what is going ...
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### How should one read the s*(t) function in Mendelson's Introduction to Mathematical Logic?

I'm self-teaching logic and doing it by means of following Elliot Mendelson's Introduction to Mathematical Logic (6th edition). In p.56 he defines a a function s* which, in his words, 'assigns to each ...
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### Countable elementary submodels

I'm having some trouble understanding elementary submodels. Let $H_\chi$ be the set of all sets which are hereditarily of cardinality $<\chi$. Let $\textbf{N}=(N,\in)$ be a countable elementary ...
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### Why in countably saturated models, types that are consistent with $TH(\mathcal{M_a})$ are finitely realizable?

I'm learning about countably saturated ($\alpha$-saturated) models. There is a hidden presupposition everywhere used: The type $\Gamma(x)$ is consistent with $TH(\mathcal{M_a})$ iff $\Gamma(x)$ is ...
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### Is model theory needed to understand ordinal logic?

By ordinal logic, I mean turing's ordinal logic. I'm going to learn first order logic, elementary set theory, basic computability, and godelian incompleteness as prerequisites for ordinal logic. But, ...
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### Let $L$ be a first order language with equality and two binary function symbols…

$\bf Question$ Let $L=(F=\{f^2,g^2\}, P=\{=\}, C=\{c\})$ be a first order language. Let $I = (\Bbb R^{2\times 2}, f^2(x,y)=(x+y)^t,g^2(x,y)=x+y,0)$ be an interpretation of $L$. Exhibit ...
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### Can't EF game theory be applied to finite languages WITH function symbols?

Let $\mathcal{M}$ and $\mathcal{N}$ be two structures in a language $\mathcal{L}$. We define the finite determined game $G_n(\mathcal{M},\mathcal{N})$ as a game with $n$ rounds where in each round ...
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### non-isomorphic countable models of $Th(\mathbb{N})$

I'm proving there are exactly $2^\omega$ non-isomorphic countable models of standard natural numbers. I got cardinality of them $\geq 2^\omega$from prime arguments. but I don't get how to prove other ...
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### Is this an axiomatization of real closed fields?

I know that real closed fields are defined as ordered fields where every positive element is a square and every odd polynomial has a root. But can they also be axiomatized as totally ordered fields ...
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### The analogy between two Rudin-Keisler orders

Given a set $X$, an ultrafilter $U$ on $X$, and a function $f\colon X\to Y$, we can push forward $U$ along $f$ to obtain an ultrafilter $f_*U$ on $Y$, defined by $C\in f_*U$ if and only if ...
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### Equivalence of the theories $\operatorname{Th}(\Bbb{R}, 0,1,+, \le)$ and $\operatorname{Th}(\Bbb{Q}, 0,1,+, \le)$

So I was working on showing that $$\operatorname{Th}(\Bbb{R}, 0,1,+, \le) = \operatorname{Th}(\Bbb{Q}, 0,1,+, \le)$$ My initial idea for working on this problem was to systematically start by ...
### Axioms for $\mathbb Z$-groups without named one?
The theory of $(\mathbb Z,+,0,1)$ has been studied as the theory of $\mathbb Z$-groups, and it has been examined as a series of exercises (and I'm sure other places) in David Marker's Model Theory ...