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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When you name an element in an uncountably categorical theory…

When you name an element in an uncountably categorical theory $T$ does it remains uncountably categorical? In other words, given a finite elementary map $f:M\to N$ between models of an uncountably ...
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Why are $\vdash$ and $\vDash$ symbols from metalanguage?

I've read in some textbooks that $\vdash$ and $\vDash$ are symbols present only in metalanguage. From this, I infer that their use in object language is unacceptable. I would like to know why. Can't ...
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47 views

Random graphs are not uncountably categorical

Is there a simple proof that the theory of random graphs is not $\lambda$-categorical for uncountable $\lambda$?
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How to prove an equality in a Lindenbaum-Tarski algebra?

Let $\mathscr{L}'= \mathscr{L}\cup \mathscr{C}$ be an extension of the language $\mathscr{L}$ with a new infinite set of constants $\mathscr{C}$, and $T$ be an $\mathscr{L}$ theory. I wish to show ...
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Why isn't second-order ZFC categorical?

This builds off of (and reuses some of the examples from) this question about failures of categoricity not related to size. Second-order $\mathsf{ZFC}$ isn't categorical, but it's almost-categorical, ...
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Proof that $\cal K$ is an elementary class iff $\cal K$ is closed under elementary substructures.

Assume $\cal K$ is a pseudo elementary class. I need to prove that $\cal K$ is an elementary class iff $\cal K$ is closed under elementary substructures. Pseudo elementary class is a class of reducts ...
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40 views

Does elementary equivalence imply L-equivalence for structure L?

Does elementary equivalence imply L-equivalence for structure L? In the textbook "A Shorter Model Theory" by Wilfrid Hodges, page 39 defines both of these terms but does not tie them together. I was ...
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118 views

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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$\kappa$-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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82 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, ...
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68 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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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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53 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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29 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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78 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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57 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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137 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
51 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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122 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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136 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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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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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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84 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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103 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
45 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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49 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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1answer
68 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
117 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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94 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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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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98 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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50 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 ...
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1answer
35 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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59 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
79 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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54 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
47 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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66 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
64 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
97 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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220 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 $L$. ...