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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Presburger arithmetic and finite model property

I'm learning about model theory and first order logic. Recently, I read about finite model property and Presburger arithmetic, and I have two questions about them: Does Presburger arithmetic has ...
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A tool to prove Baby Ax-Kochen principle

I have been reading a lecture notes on model theory of valued fields written by Lou van den Dries. This is the question I have: It is known in this lecture that: Let $R$ be a local ring, with $t\in ...
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preservation in unions of chains

Let $K=\{A_i:i\in\omega\}$ be a countable chain of infinite (not necessarily countable) N−substructures, where N is a binary relation and let A be the limit (union) of K. Let Ax be a $\Pi_2$ ...
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Proof applying compactness theorem.

I am trying to work out the next proof: Let $\Sigma$ be a set of formulas. Assume $$\Sigma \vDash \bigvee_{i \in I} \varphi_i \ \leftrightarrow \ \bigwedge_{j \in J} \psi_j $$ Where $\varphi_i$ and ...
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76 views

About the proof of a test for quantifier elimination.

I've been reading D. Marker's book on Model theory. In the part dealing with quantifier elimination there's a corollary I've been trying to prove without any luck: Corollary 3.1.6 Let $T$ be an ...
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prove that the class of cyclic groups is not axiomatizable?

1)prove that the class of finite groups is not axiomatizable? Suppose there is a set $\Sigma$ of first-order sentences such that $\mathrm{Mod}(\Sigma)$ is the class of all cyclic groups. and how to ...
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stratification (typage) of logic and syntax at the same time: is such a dream feasible? [closed]

This post is more philosophical than formal, yet I think it's an important question. There's an idea I have for long times already that would consist, in some sense, in doing a "theory of theories". ...
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Question about the proof of Theorem 3.1.4 in Marker's Model Theory, An Introduction

On page 73 of Marker's Model Theory, An Introduction the following theorem can be found: Theorem 3.1.4 Suppose that $L$ contains a constant symbol $c$, $T$ is an $L$-theory, and $\varphi ( \bar v)$ ...
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Axiomatisability of the class of finite groups

I'm trying to prove that the class of all finite groups is not finitely axiomatizable. I was trying to do this in the same way one proves that the class of finite sets is not axiomatizable, i.e. Let ...
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Let $\alpha$ be any cardinal. There are at most $2^{\alpha \cup ||\mathscr{L}||}$ nonisomorphic models for $\mathscr{L}$ of power $\alpha$

This is an exercise from Chang & Keisler, specifically, exercise 1.3.1, though I'd also like some information on the (I think) related exercises 1.3.6 and 1.3.8. It's as the title of this question ...
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Is type definability always closed under existential quantification?

Fix a language $L$, a model $M$ and a set $A\subseteq M$. Let $p(x,y)\subseteq L(A)$ be a type. If $M$ is saturated and $|A|<|M|$ then for every $a\in M$ $$M\models \exists y\ ...
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substructures, superstructures, and their elementary counterparts

Suppose we have structures $M \subseteq N \subseteq P$ in some first order language. If $M \prec N$ and $M \prec P$, does it follow that $N \prec P$? If not, what is a counterexample?
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Underlying Set in Model Theory

In model theory a structure has an underlying set. In addition to the interpreted relations, are there (implicit) assumptions made about possible operations on this set? For example, is it assumed to ...
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52 views

Can the empty-set be used for a model that satisfies an axiom system?

I'm reading some notes on Model Theory and wondering if axiom systems that make no existential claims are trivially satisfied by the empty set. For instance, if you just have the axiom that all ...
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Please unpack this notation from book Model Theory by Marker

On page 73 of Model Theory by Marker, he proves DLO has quantifier elimination. In it he writes: For $\sigma: \{(i,j) : 1 \le i < j \le n\} \rightarrow 3$, let $\chi_{\sigma}(x_1,\ldots,x_n)$ be ...
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41 views

Proving this axiom system is consistent.

I want to make sure I understand the correct notation and expressions for proving that an axiom system is consistent. I have the axioms Every line is a collection of points. There are at least two ...
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Finitary assignment functions and typed modal languages

I'm working through Giovanna Corsi's article on Counterpart Semantics for Modal Logic. She is working with a typed modal language $\mathscr{L}^{t}$. Where this differs from usual presentations I've ...
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33 views

Question about infinitary logic

Let $\mathcal{L}$ be the language based on countably many sentence letters $p_i$, $\neg$, and infinite conjunction $\bigwedge$. Say that $p_i$ is an essential constituent of an $\mathcal{L}$-sentence ...
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Elementary equivalence of $\mathbb{Z}^{n}$

I have a question in model theory where I need to prove that $\mathbb{Z}^{n} \not\equiv \mathbb{Z}^{m}$ (= not elementarily equivalent) if $n\neq m$ in the language of groups $L = \{\circ ,^{-1},0 ...
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In what sense of “structure” do group homomorphisms “preserve structure”?

It is commonly said that group homomorphisms "preserve the structure of the group", e.g., from Wikipedia: The purpose of defining a group homomorphism as it is, is to create functions that ...
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Predicate Calculus Logic Models, Exam revision I simply don't understand.

I am studying for an exam to be taken next Friday and I am only stuck on one question in my revision guide, which I simply do not understand. (I even have the answers as well) The question: Show, ...
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68 views

What are non-categorical theories about?

With theories that are categorical, it seems like you could say that the theory is about collections of objects (numbers, points, etc.) with a certain structure (the structure the standard models ...
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77 views

“Induction on Complexity” Problem

Let $F,G$ be two distinct atomic formulae. Show that no formula that can be formed from $F,G$ using just the conditional $\to$ is logically equivalent to $\neg F \vee \neg G$. We are then told that we ...
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“Equivalence” of types in model theory

Unfortunately, my question is a bit vague. Since I was not able to find anything about it in the literature, I decided to ask it on this forum. Here it is: Let $A$ be a model in some language $L$. ...
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Show if theory T is ∀∃-axiomatizable, then T has an existentially closed model.

Setting Definition $\mathcal{M} \models T$ is existentially closed if whenever $\mathcal{N} \models T$, $\mathcal{N} \supseteq \mathcal{M}$, and $\mathcal{N}\models \exists \bar{v} ...
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Is definable $\emptyset$-definable?

A set $A$ is $X$-definable in an $L$-structure $\mathcal{M}$ with a domain $M$ iff there are an $L$ formula $\phi(a,\bar x)$ and $\bar x \in X^n$ such that $A=\{a \in M^m | \mathcal{M} \models ...
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77 views

If T is forall/exists-axiomatizable and M,N satisfies T, then there is exists/forall sentence $\psi$ so that If M sat $\psi$, then N sat $\psi$

Setting Definition: A theory $\pmb{T}$ has a $\forall\exists$-axiomatization if it can be axiomatized by sentences of the form $$\forall v_1\ldots \forall v_n \exists w_1 \ldots \exists w_n ~~ ...
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Prove this proposition concerning a theory with ∀∃-axiomatization part II.

This is a continuation of a problem I asked yesterday seen here: Prove this proposition concerning a theory with ∀∃-axiomatization The setting is reproduced below for easy reference. Setting A ...
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Tarski-Vaught test, unclear implication

We know that Tarski-Vaught criterion says: $M$ is an elementary submodel of $N$ iff $M$ is a submodel of $N$ and when $\overline{a}\in|M|^n$, $b\in |N|$, $N \models\phi[b,\overline{a}]$, then there is ...
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If you create a new theory of a model by augmenting the language with constants, does the new theory contain sentences that says anything “original”?

Suppose $\mathcal{M} \models T$ where $T$ is a set of $\mathcal{L}$-sentences. Now create an augmented language by adding all the constants in $\mathcal{M}$: $$\mathcal{L}^* = \mathcal{L} \cup \{c : ...
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86 views

Prove this proposition concerning a theory with ∀∃-axiomatization

Setting A theory $\pmb{T}$ has a $\forall\exists$-axiomatization if it can be axiomatized by sentences of the form $$\forall v_1\ldots \forall v_n \exists w_1 \ldots \exists w_n ~~ ...
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Covariant and countervariant interpretations of SEQ in AR

Let a covariant interpretation of model $\mathscr{M}$ in model $\mathscr{M}'$ be defined as a couple of functions $(f,g)$, with $f$ injective, such that ...
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What can we conclude if $\mathcal{N}$ is an elementary extension of $\mathcal{M}$?

In particular, if $\mathcal{N}$ is an elementary extension of $\mathcal{M}$, can we immediately conclude that if $\mathcal{M} \models T$, then $\mathcal{N} \models T$? Why or why not? My line of ...
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Check my proof of “overspill” in non-standard models of Peano arithmetic.

Proposition Let $\mathcal{M}$ be a nonstandard model of Peano arithmetic, $\phi(v,\bar{w})$ a formula in the language of arithmetic, and $\bar{a} \in \mathbb{M}$. Show that if $\mathcal{M} \models ...
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Definitions of isomorphism and elementary substructures

Let us define -all the definitions are from V. Manca, Logica matematica, 2001, 'mathematical logic'- a $\Sigma$-morphism as a function $f:D_1\to D_2$ between models $\mathscr{M}_1$ and ...
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Show that the subset of a real ordered field defined by a ring formula has a least upper bound.

So currently trying to get well practiced in model theory, and i have come across the following question which i need some help with. Esentially let $S \subseteq \mathbb{R}$ be a non empty set, ...
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Show that any model of $\Delta$ is a Nonstandard Model of Arithmetic

I was hoping that someone could help check my proof. I originally thought I was spot of with my proof, but my professor suggested that my method was wrong. So, I went to check the hint in the back of ...
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Two Place Position and Model Question

! i get trouble in one multiple choice question in logic course: any one could help me with some description ? if we have Two-place position predicate, like : 1) all models of $\varphi$ is ...
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Find a formula that separate between structure

I have language $ L = \{ < \} $. I have the following structures: $|M| = \{ 1-\frac{1}{m} |m\in Z, m >1\} $ $|N| = \{ 1-\frac{1}{m} - \frac{1}{n} |m,n\in Z, m,n >1\} $ I need to find a ...
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Could someone give a concrete example of an amalgamation from model theory?

The theorem I have in mind can be found in page 135 of this book: ...
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If $\nu : \mathcal{M} \rightarrow \mathcal{N}$ is an elementary embedding, then is it true that $\mathbb{M} \subseteq \mathbb{N}$?

Exactly as the title suggests: If $\nu : \mathcal{M} \rightarrow \mathcal{N}$ is an elementary embedding, then is it true that $\mathbb{M} \subseteq \mathbb{N}$?
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what does the “fixed point” in fixpoint algorithms refer to?

I was reading the following powerpoint here to remember something I studied a long time ago. http://www.cs.cmu.edu/~emc/15-820A/reading/lecture_1.pdf the 12th slide is labeled fixpoint algorithms, ...
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Proving amalgamation property in model theory

Restate the proposition Suppose $\mathcal{M}_0$, $\mathcal{M}_1$, and $\mathcal{M}_2$ are $\mathcal{L}$-structures and $j_i ~:~ \mathcal{M}_0 \rightarrow \mathcal{M}_i, ~(i = 1,2)$ is an elementary ...
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Using first order sentences, how would you write down the axioms of finite field?

Suppose my language is $\mathcal{L} = \{+,-,\cdot,0,1\}$. My attempt is as follows: $$ \forall x \forall y \forall z (x - y = z \leftrightarrow x = y + z)\\ \forall x x \cdot 0 = 0\\ ...
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90 views

Prove elementary amalgamation theorem in model theory.

Restate the proposition Suppose $\mathcal{M}_0$, $\mathcal{M}_1$, and $\mathcal{M}_2$ are $\mathcal{L}$-structures and $j_i ~:~ \mathcal{M}_0 \rightarrow \mathcal{M}_i, ~(i = 1,2)$ is an elementary ...
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Examples of forcings which add no “definable” Aronjain tree

Maybe a bit board question but: Fixing a regular cardinal $\kappa$ in the ground model, I am looking for examples of set forcing notions which preserve regularity of $\kappa$ and add no new $\kappa$ ...
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Suppose some theory T has countably many axioms, how many models of $T$ are there of cardinality $\aleph_1$,$\aleph_2$,$\aleph_{\omega_1}$?

Setting Let $\mathcal{L} = \{E\}$ where $E$ is a binary relation symbol. Let $T$ be the $\mathcal{L}$-theory of an equivalence relation with infinitely many infinite classes. So we see $T$ has ...
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Using first order sentences, axiomize the $\mathcal{L}$-theory of an equivalence relation with infinitely many infinite classes.

Problem Let $\mathcal{L} = \{E\}$ where $E$ is a binary relation symbol. Let $T$ be the $\mathcal{L}$-theory of an equivalence relation with infinitely many infinite classes. Current Solution ...
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How would you define the set of even numbers in $\mathbb{Z}$ using a first order sentence?

Given a language $\mathcal{L} = \{+,0\}$ and structure $\mathcal{M}$ with underlying universe $\mathbb{Z}$, I would like to write a formula or sentence $\phi$ so that the set $$\Big\{ \bar{a} ~:~ ...
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Domain of functions interpretated in model

I read (V. Manca, Logica matematica, 2011) that the set $T_\Sigma(V)$ of the terms on a signature $\Sigma$ and variables $V$ is inductively defined letting, for any generic variable $x$, ...