# 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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### Number of equivalence classes based on a relation regarding a non-principal ultrafilter

We have an equivalence relation on $\mathbb{N}^\mathbb{N}$ given by $$f\equiv g \iff \{n\in\mathbb{N}: f(n)=g(n)\}\in\mathbb{U},$$ where $\mathbb{U}$ is a non-principal ultrafilter on $\mathbb{N}$. ...
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### Can someone explain to me this logic sentence using entailment?

Can someone please explain me what does it exactly mean? $KB \wedge B^- \not\models\square$ I understand the entailment symbol in this example here : $T \models A$ is if there's no model of $FS$ ...
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### easy proof of the completeness theorem [closed]

The completeness theorem of first-order logic states: If $\Phi\models\phi$, then $\Phi\vdash\phi$. Assume that I have a calculus $\vdash$ in mind for which I want to prove this completeness theorem. ...
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### Isomorphism, Model-Theorie, “beginning”

Let $\mathcal{N}^{\ast}=(N, <_{\mathcal{N}^{\ast}},...)$ be a model of $Q$ (the Robinson arithmetic). Show, that it exists a "beginning" (definition below) of $\mathcal{N}^{\ast}$ which is ...
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### Peano Arithmetic, proof

Show in PA: $\forall v_0\forall v_1 (v_0<v_1\rightarrow \exists v_2\quad v_0+v_2=v_1)$ Hello, I have a question to this task, because I do not know, how to proof this. I give the definition ...
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### Would a theory with model $\{\}$ be consistent?

I don't know much about model theory, but it was suggested in another math forum that a theory with model $\{ \}$ would be consistent. Is this correct?
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### Compactness theorem

Let $\Sigma$ be a set of theorems, such that for every $\varphi\in\Sigma$ exists an arbitrarily large (<--- edited) finite modell $\mathcal{M}$, with $\mathcal{M}\models\varphi$. Show: It ...
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### Is it circular to define the Von Neumann universe using “sets”?

I was just reading the Wikipedia page on the Von Neumann universe, where it is stated that this universe "is often used to provide an interpretation or motivation of the axioms of ZFC." However, later ...
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### Explicit example of countable transitive model of $\sf ZF$

Do we know any explicit example of a countable transitive model for $\sf ZF$ or $\sf ZFC$?
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### Tarski's schema T

On Wikipedia, Tarski schema T says: A sentence of the form "A and B" is true if and only if A is true and B is true A sentence of the form "A or B" is true if and only if A is true or B is true A ...
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### Generators of the Lindenbaum-Tarski algebra

I am a bit confused about the role of propositional variables in the construction of the free Lindenbaum-Tarski algebra. In the entry "Lindenbaum-Tarski algebra" on Wikipedia, in the section "...
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### Elimination of quantifiers for the theory of equivalence relations with two infinite classes by back-and-forth

As I said in an earlier question, I'm trying to understand how to obtain elimination sets by way of back-and-forth arguments. Since I'm not totally sure I understood how it works, I wanted to check my ...
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### Graded back-and-forth systems and unnested Ehrenfeucht-Fraïssé games

I'm trying to work my way through back-and-forth systems and elimination sets by reading the relevant sections in Hodges' Model Theory and I'm a bit confused by one of his lemmas (specifically, it's ...
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### Sum of measurable functions is measurable: countable choice required?

The standard proof that the sum of measurable functions is measurable uses countable choice, via the countable subadditivity of outer measure ($\implies$ measurable sets are closed under countable ...
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### Inclusion in sequences of theories and models

Is this theorem true, and if so does it have a name and where can I reference it? "Let $T_1$ and $T_2$ be theories where $T_1 \subset T_2$. If $K_1$ and $K_2$ are the classes of all models of $T_1$ ...
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### Cardinality of Galois groups

We know that no Galois group of a Galois extension is countable. The question is: which cardinalities are possible for a Galois group? (or also: for profinite groups?) I suspect that the theory of ...
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### Do the axioms of set theory actually define the notion of a set?

In Henning Makholm's answer to the question, When does the set enter set theory?, he states: In axiomatic set theory, the axioms themselves are the definition of the notion of a set: A set is ...
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### Finding Model Containing Commutative Diagram as Elementary Submodels

I'm looking at a justification for why we work inside a big model, but I'm having trouble proving a particular comment. Assume we're working with a complete theory $T$ and we have a commutative ...
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### Is it possible that Gödel's completeness theorem could fail constructively?

Gödel's completeness theorem says that for any first order theory $F$, the statements derivable from $F$ are precisely those that hold in all models of $F$. Thus, it is not possible to have a theorem ...
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### Quantifier elimination, $(\mathbb{R}, <)$

I have a general question about quantifier elimination. Which kinds of formulas do you have to observe? For example let T be the theory of $(\mathbb{R}, <)$ and I want to show, that this theory ...
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### Why does a finite set having a model imply that the set is consistent [closed]

Assuming the soundness theorem to be true, can someone explain why if we assume $\Sigma$ has a model $M$. Then $\Sigma$ is consistent ?
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### Characterizing coding with automorphisms

I am attempting the following exercise from chapter 5 of Van den Dries' notes "Introduction to Model-Theoretic Stability". I suspect the exercise shouldn't be too difficult but I've become pretty ...
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### Definable over $(\mathbb{N},0,S)$, $A$ or $\mathbb{N}\setminus A$ is finite

Let $A\subseteq\mathbb{N}$ be defineable over $(\mathbb{N},0,S)$. Then is $A$ or $\mathbb{N}\setminus A$ finite. $S$ is here the successor-function $S(n)=n+1$. Hello, I have a question to this task. ...
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### Gödel's ontological proof and “modal collapses”

Recent findings on Gödel's ontological argument allowed to ultimately establish a couple of things: Gödel's original axiomata are inconsistent Scott's variation instead is consistent Scott's axioms ...
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### “finally cyclic”, definable over $(\mathbb{N},+)$

Let $A\subset\mathbb{N}$ "finally cyclic" (I give the definition below). Show, that $A$ is definable over $(\mathbb{N},+)$ Hello, I have a question to this task. I have to show, that the set $A$ is ...
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### Does the theory of equivalence relations have quantifier elimination?

I am aware that the theory of equivalence relations with infinitely many classes, all of which infinite, has quantifier elimination, as can be seen from the answer to this question. However, does the ...