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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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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55 views

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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32 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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1answer
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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73 views

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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71 views

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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2answers
53 views

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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59 views

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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1answer
85 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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55 views

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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38 views

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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2answers
85 views

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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73 views

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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1answer
59 views

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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1answer
37 views

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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1answer
51 views

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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1answer
39 views

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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40 views

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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66 views

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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1answer
53 views

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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1answer
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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70 views

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 $\kappa$ ...
4
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1answer
82 views

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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1answer
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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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31 views

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 $x$, ...
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Is $\mathbb{N}$ definable in $(\mathbb{Z},0,S)$?

Is the subset of natural numbers (first-order) definable in the structure $(\mathbb{Z},0,S)$ ($S$ is the successor function)? I believe that it cannot, since $\mathbb{N}$ is exactly the set of ...
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2answers
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Examine my argument that $\mathbb{Z} \oplus \mathbb{Z} \not\equiv \mathbb{Z}$

Setting Let $\mathcal{L} = \{+,0\}$. I want to show that $$\mathbb{Z} \oplus \mathbb{Z} \not\equiv \mathbb{Z}$$. Note $\equiv$ means elementary equivalence in this question. Updated Problem My issue ...
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91 views

A proof of the Löwenheim-Skolem-Tarski theorem

The Löwenheim-Skolem-Tarski theorem, which says that If a theory has a model of a given infinite cardinality, then it has models of any greater infinite cardinality is proved by my textbook in the ...
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81 views

A first order sentence such that the finite spectrum of that sentence is the following subsets of $\mathbb{N}^+$.

I would like to find a language $\mathcal{L}$ and first order sentence $\phi$ of $\mathcal{L}$ so that its finite spectrum is $\{p^n ~:~ n > 0, \text{ p is prime}\}$ $\{ p ~:~ p \text{ is ...
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1answer
48 views

Show some $\mathbb{X} \subseteq \mathbb{N}^+$ occurs as the finite spectrum of a sentence for this language.

Setting Define the finite spectrum of an $\mathcal{L}$-sentence $\phi$ as $$\{ n \in \mathbb{N}^+ ~:~ there ~ is~ \mathcal{M} \models \phi ~with~ |\mathbb{M}| = n\}$$ And let $$\mathbb{X} = \{ 2^n ...
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Walk me through the proof that the class of graphs is elementary.

I am attempting problem 1.4.5 in "Model Theory" by Marker, but I am very lost in terms of how to get started. So to help me with the process I would like someone to show me that the class of graphs is ...
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108 views

Distinguish between substructure, submodel, elementary substructure, and elementary submodel.

I can see (although I must not really understand) the definition of these terms, but could someone please explain the difference between these concepts, and whether any one of them imply the other? ...
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How to calculate the cardinality of a model

I know, thanks to some clarifications received from a user of this site, the definition of a model. When evaluating the cardinality of a model by taking the interpretations of all the constants, ...
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1answer
64 views

What is finite in a finite model

I am studying some theorems of model theory in an introductory text of mathematical logic. I know that a model is a way of associating the relationary symbols of a signature $\Sigma$ to $k$-ary ...
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No proposition $\chi$ such that $\mathscr{M}\models\chi\iff\mathscr{M}$ is infinite

Let notation "$\models$" be used for the two following case: let $\mathscr{M}\models\varphi$, where $\mathscr{M}$ is an interpretation model and $\varphi$ is a proposition, mean that $\varphi$ holds ...
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Definable sets, substructures and unions

Ok so my question is as follows; Let A be a substructure of B and S $\subset$ B be a definable set define by a universal formula $\phi(x)$, I need to show that in A $\phi$ defines $A \cap S$ My ...
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1answer
78 views

L-sentence which expresses bijective function

I've stumbled upon this exercise from "Sets, Models, Proofs" and can't seem to find a solution. It goes like this: Let $L$ be a language with just one 1-place function symbol $F$. Give an ...
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Possible not countable extension of the natural numbers?

This question comes from:Is $1234567891011121314151617181920212223......$ an integer? We define $\mathcal{A}$ as the set of infinite strings of digits $$ \bar a_i=a_0 a_1a_2a_3\cdots a_i \cdots ...
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1answer
42 views

What is an omega model?

I went to a seminar and a side question was if a theory had an omega model, however from the context I could not deduce the exact meaning. Does an omega model have a general meaning in mathematical ...
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59 views

Show this language structure models this sentence.

In an effort to educate myself, I am attempting the second problem in first chapter of the book "Model Theory" by Marker. The problem is reproduced below: Let $\mathcal{L} = \{\cdot, e\}$ be the ...