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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Very Simple Model Theory

I'm working through the fifth edition of Dirk van Dalen's 'Logic and Structure' and got stuck in section 4.3 on model theory. Let a structure (of some type) be a tuple $ \mathfrak{A} = (A; R_1, ...
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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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58 views

$\kappa$-saturated, $1$-types - $n$-types

Definition. Let $\kappa$ be an infinite cardinal. We say that an $L$-structure $\mathfrak{A}$ is $\kappa$-saturated iff all $1$-types over sets of cardinality less than $\kappa$ are realised in ...
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No countable models

I want an example of a theory T with finite models of arbitrarily large size but T has no countably infinite model. I know that T has to be uncountable, but couldn't come up with an example. ...
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65 views

Failures of categoricity not related to size?

The paradigmatic cases I know of theories failing to be categorical are cases that seem tied to the language's inability to distinguish between different infinite cardinalities of the theories domain. ...
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Show there is no elementary extension of $\mathbb{N}$ with an element between $0$ and 1

I have been presented with the follwing question and i want to see if the method i have used works, i have my doubts. We recall that M is an elementary extension of $N= \langle \mathbb{N}; +, ., 0, 1 ...
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52 views

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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20 views

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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43 views

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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27 views

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 ...
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1answer
66 views

Non Archimedean countable models of the theory of the reals

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 ...
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50 views

Walk me through this proof that a theory is satisfiable.

Setting Suppose $\mathcal M, \mathcal N \models T$, $\mathcal M \subseteq \mathcal N$, $\mathcal M$ existentially closed, then I I want to prove that there is $\mathcal M_1 \models T$ so that ...
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60 views

Logical Structure of a Proposition

I'm having a hard time figuring out the logical structure of the following theorem : I'm not interested in proving it, for now, i'm just trying to understand its logical structure. I don't know ...
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1answer
42 views

Independence Property in Model Theory

Often, the Independence Property is defined in the monster model of a complete theory. When it is not, it usually goes like: a formula $\phi(x, y)$ is said to have the independence property if for ...
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2answers
105 views

Definable subsets of $\mathbb{Q}^2$ in $\langle \mathbb{Q} , < \rangle$?

The question seems quite simple; what are the definable subsets of $\mathbb{Q}^2$ over the structure $\langle \mathbb{Q} , < \rangle$. Part of me wants to say there are none, given any definable ...
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96 views

Models of the empty theory

so throughout my reading of model theory the idea of the "empty" theory has been put down as trivial, however I am curious as to why. Let us look at the following. Suppose We have $L_=$, the language ...
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38 views

Algebraic characterization of first order “operations” using limit ultrapowers

In his review http://projecteuclid.org/download/pdf_1/euclid.bams/1183537899 of the Chang and Keisler's classic book "Model Theory," M. Makkai writes: "… let us note that it is possible to formulate ...
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60 views

Open interpretation of logical theories

This may be more appropriate for MO but I thought I'd ask here first as it's just a question about logic (not my strong point at all but not research-level in itself). I'm going through Razborov's ...
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80 views

Skolem functions in the real ordered field.

I am currently reading into a bit of model theory and have come across the idea of Skolem functions, as used in the proof of the downward Lowenheim-Skolem theorem. Despite seeing their use there I ...
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75 views

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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1answer
36 views

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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39 views

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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60 views

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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1answer
80 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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62 views

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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69 views

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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66 views

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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1answer
42 views

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 ...
3
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1answer
29 views

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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41 views

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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1answer
55 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 ...
2
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2answers
50 views

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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1answer
42 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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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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1answer
41 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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79 views

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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6answers
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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 ...
2
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1answer
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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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86 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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75 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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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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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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1answer
65 views

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
54 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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1answer
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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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 ~~ ...