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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On the existence of finite substructures when sufficient chain conditions are met

Let $L$ be a language and $T$ and $L$ theory. Suppose that for any $M\models{T}$, we have $M\subseteq{\bigcup{C_{n}}}$, where each $C_{n}\models{T_{\forall}}$ is finite. I want to show that for some ...
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1-model complete

For $L$ structures $A$ and $B$ we write $A\preceq_{1}B$ if $A\subseteq{B}$ and $A\models{\varphi(a)}$ iff $B\models{\varphi(a)}$ for any finite tuple (of the correct length) $a$ from $A$ and for any ...
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110 views

Class models of $\mathsf{ZFC}$ and consistency results

First of all, I'm only starting to study independence results in set theory. And there is one obstacle that confuses me a lot. Probably such questions have already been asked, but I haven't found ...
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65 views

Indiscernibles over a model

Working within the framework of a monster model, I wish to show that: (*) If $(a_{i}:i<{\lambda})$ is an indiscernible sequence over $A$, then there is a model $M$ containing $A$ such that $(a_{i}:...
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39 views

Definition of Local Finiteness

Let $L$ be a language and let $T$ be an $L$-Theory. $M\models{T}$ is said to be locally finite if for any given finite subset $X$ of $M$, there is a finite substructure $A$ of $M$ s.t. $X\subseteq{M}$....
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55 views

D.Marker's axiomatization of rings

Adding "-" as a binary function to the language of rings and the sentence $∀x(x+(−x)=0)∀x(x+(−x)=0)$ to the set of axioms proves existence of additive inverses. But I can't see how Professor Marker's ...
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227 views

Upward Löwenheim Skolem

I'm trying to understand the proof of (a version) of the upward Löwenheim Skolem Theorem, which states that given a language $\mathscr{L}$ and a set of $\mathscr{L}$-sentences $\Sigma$ with a ...
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72 views

Absoluteness and Extensionality

In the set theory text that I am reading, the author writes: Relative to the set $A = \{ 0, \{\{0\}\} \}$, the sets $0$ and $\{\{0\}\}$ are indistinguishable in the sense that $[$for all $x$ in $A$...
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61 views

elementary equivalence and incompleteness

I read the following line in a text on set theory: "Peano Arithmetic has continuum many non-isomorphic countable models (including the standard model omega), all of them elementary equivalent." ...
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89 views

ACF universal is the theory of integral domains

When studying David Marker's "Model Theory: An Introduction" book trying to understand the proof of Lemma 3.2.1 which says: $ACF_{\forall}$ is the theory of integral domains, I couldn't understand the ...
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54 views

A test for quantifier eliumination

In David Marker's "Model Theory: An Introduction" book I was trying to prove Corollary 3.1.12 which is left for the reader, but I couldn't reach any solution. The aforementioned corollary states: ...
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59 views

An AE axiomatization of groups

Let $L=\{*\}$. The usual axiomatization of groups in this language has the EA axiom $\exists{e}\forall{x}$ $ e*x = x$. But the union of a chain of groups is also a group. This means that the theory of ...
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Shelah's materialize vs. realize;note: tags are badly chosen due to the lack of them

Can someone please explain to me in some detail the exact difference between materialize and realize for a Galois type $p$? Esp. is realize a special case for materialize? Why is it so? What is the ...
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71 views

Clarify definitions of relation and 0-ary relation

From mathworld.wolfram.com: A relation is any subset of a Cartesian product But if so, then the null set is all of: 0-ary, 1-ary, 2-ary etc. Wouldn't it be better to define it as: A relation ...
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92 views

Definition for non-dividing

The definition for non-dividing is taken as the negation of the definition for dividing (as found in http://www.math.cmu.edu/~rami/simple.pdf : Definition 1.1 for example). Thus assuming $\varphi(x,c)$...
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62 views

When can independence of a statement in a theory be reduced to “truth”?

Since the Goldbach conjecture is in $\Pi_1^0$, if it were proven to be independent of Peano Arithmetic, it would follow that the Goldbach conjecture is true (i.e. true in the standard model), since ...
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47 views

Prove that the Morley Rank is preserved under definable bijections.

I need to prove this: If there is a definable bijection between $\varphi(C)$ and $\psi(C)$ then $RM(\varphi)= RM(\psi)$. Where $C$ is the monster model. I can intuitively understand it, the Morley ...
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91 views

prove that any two isomorphic structures are elementarily equivalent

Imagine we have two L-structures $M$ and $N$. For each L-sentence $\phi$ , $M$ models $\phi$ iff $N$ models $\phi$. We call $M$ and $N$ two elementary equivalent L-structures. We say $M$ and $N$ ...
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78 views

Understanding types and the proof that every type is realized in an elementary extension.

So I've been recently been studying types from David Marker's book and have some issues understanding them and in particular why did Marker choose to present certain proof of the following theorem ...
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70 views

Type of Infinite Tuple

Things along the following lines is often said about infinite tuples in model theory (we are assuming that we are working inside some monster model $M$ of some complete $L$ theory $T$): If I is a ...
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31 views

Is there a conservative extension of IZF that extends IZF by a weak form of the axiom of choice?

The full axiom of choice implies the LEM, and so is incompatible with constructive mathematics, although there are some weaker forms of the axiom of choice, such as the axiom of dependent choice, or ...
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44 views

Forking in Strongly Minimal Theories

I have been trying to define $A\overset{\vert}{\smile}_{C}B$ in a strongly minimal theory (let's say countable to avoid complications though I'm not sure if this matters). My attempt is based on the (...
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554 views

Intervals in divisible ordered groups

Is it true that if $(G,+,0,<)$ is a divisible ordered abelian group with at least two elements, then for $a,b >0 \in G$, there is an injective order preserving map from $[0;a)$ to $[0;b)$? It ...
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52 views

Representing a $\sigma$ - structure using a signature-$\sigma$ in Mathematical Logic.

In mathematical logic, I have a question regarding how a signature-$\sigma$ relates to a corresponding $\sigma$ structure which interprets the signature-$\sigma$ In Chiswell and Hodges book "...
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Explicit countable elementary extension of $\mathbb{N}$

I would like to see an explicit example of a non-trivial elementary extension of the structure $(\mathbb{N}, +, \cdot, 0, 1)$ where $\mathbb{N}$ includes zero. Most of all I am interested in countable ...
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87 views

How does Gödel's second incompleteness apply to any theory containing arithmetic?

If I understand correctly, there are two facts proven by Gödel's second incompleteness theorem, for a formal theory containing arithmetic 1) It is possible to express the consistency of the theory ...
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40 views

To build a tree in a countable theory with uncountable many types in finite variables

I'm trying to prove that if $T$ is a countable consistent theory that has no binary trees, then $T$ is small. i.e $|S_n (T)|=\aleph _0$ In order to do this i assume toward contradiction that $S_n (T)...
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51 views

If $\mathcal{T}_1$ and $\mathcal{T}_2$ admit quantifier elimination, does $\mathcal{T}$ admit quantifier elimination?

Let $\mathcal{T}_1$ and $\mathcal{T}_2$ be theories with disjoint signatures $\mathcal{L}_1, \mathcal{L}_2$. Form a new language $\mathcal{L} = \mathcal{L}_1 \cup \mathcal{L}_2 \cup \{P_1, P_2\}$, ...
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91 views

How do we know $\mathbb{N}$ is a model of Peano Arithmetic?

The induction axiom in the theory of Peano Arithmetic (PA) is actually an axiom scheme such that for every formula $\phi(x,\bar{y})$ with free variables $x,\bar{y}$ ($\bar{y}$ being a string of ...
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Hodges exercise 2.7.1: Quantifier elimination in dense linear orderings

In Hodges' A Shorter Model Theory, exercise 2.7.1 tells you to prove theorem 2.7.1, which says that the following five formulas are an elimination $\Phi$ set for the class of all dense linear ...
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36 views

two different realization of one type

Im having trouble solving this problem: Let $M$ be a saturated structure of cardinality $\kappa$. Let $A\subseteq |M|$ with $|A|<\kappa$. Then there is a type $p\in S_1(A)$ with two different ...
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What does the theory of the empty set look like?

I know next to nothing about logic, but I was wondering what first order axioms would give rise to the theory of the empty set (that is to a theory whose only model is the empty set)? The problem I ...
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140 views

Detecting incomparability in countable elementary submodel

This might be just an easy exercise in model theory but I can't seem to wrap my head around right now. Let $\theta$ be large enough regular cardinal and $\kappa < \theta$. $(\kappa, \prec)$ is ...
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58 views

Quantifier elimination in infinitary languages

Let $L$ be a first order relational language with identity. Let $E$ be an $L$-structure. Let $L_{\alpha\beta}$ be the usual infinitary extension of $L$. Thus, $E$ is an $L_{\alpha\beta}$-structure ...
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22 views

satisfaction of a set in the expanded language

Consider a set $\Gamma$ of formula which is satisfiable in language $\mathcal{L}$. let $\mathcal{L'} \supseteq \mathcal{L}$ be any expansion. then is $\Gamma$ satisfiable in $\mathcal{L'}$? I can ...
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90 views

Elimination of quantifiers

What does it mean that a theory admits constructive elimination of quantifiers? A theory admits elimination of quantifiers when each formula of the theory is equivalent to a quanifier-free formula, ...
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55 views

bound and free variables

I have a question that has been bothering me for quite some time. In second order logic sometimes there is an indication that a variable can be both bound and free. The simplest example I can give is ...
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63 views

Why can't an $\omega$-stable theory have finitely many countable models?

This is a "well-known fact," but I'm at a loss to finding a proof. I could swear I've read it somewhere, but checking the handful of places I'm used to checking doesn't help. Google gives nothing, ...
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56 views

Prove that $K$ is finitely definable iff $K$ has finite support

Hi guys I need to prove a Finite Support Theorem which states that $K$ is finitely definable iff $K$ has finite support. Unfortunately I succeeded in proving only the first part of if and only if. $...
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52 views

Different models of complete theory modeled by ($\mathbb{N}$, $\leq$)

This is a homework excercise. I have a FO language $\mathcal{L}$ without identity ($"="$) with a binary predicate symbol $P$, an $\mathcal{L}$-structure $\mathcal{N}$ with ground set $ \mathbb{N}$ ...
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185 views

Models in set theory and continuum hypothesis

Some days ago I had the opportunity to listen to the talk about model theory and connections with algebra and geometry. I'm not at all specialist in this field so my question probably will be naive ...
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43 views

Prove that any countably incomplete ultraproduct of a collection of models is $\aleph_1$-saturated

I'm using this article for the proof. everything sounds well, but I don't think I have a proper comprehension on some (especially the final) parts. For example, what is the function $f$ doing at the ...
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48 views

Discussion on AC and countable unions of countable sets

My question stems from the comments following Asaf Karagila's answer here : Why can't you pick socks using coin flips? At some point there was a discussion about whether or not a countable union ...
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Is the number of orbits of the automorphism group of infinite field with a finite characteristic acting of the field is finite?

I am trying to solve some statement in Model theory. And if i can show that given an infinite field $\mathbb{F}$ with a finite character, then the number of orbits of $Aut(\mathbb{F})$ acting on $\...
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What's wrong with my proof on “any countably incomplete ultraproduct of a collection of models is $\aleph_1$-saturated”?

I'm using this article for the proof. I thought some parts are extra and tried to make a new shorter proof. Here goes: Let $\Delta(x)$ be a set of formulas (with one free-variable $x$) in the ...
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Why is there a need for ordinal analysis?

Consider the Peano axioms. There exists a model for them (namely, the natural numbers with a ordering relation $<$, binary function $+$, and constant term $0$). Therefore, by the model existence ...
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25 views

End Extension models of $I\Delta_0$

Recently I'm thinking about below question, but I can not prove or disprove it. Is it true that for every model $M\models I\Delta_0$ there exists a model $M'\models PA$ such that $M'$ is end ...
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69 views

Tarski-Vaught Test - why only one variable tested at a time?

The book Model Theory: An Introduction by David Marker states the Tarski-Vaught test for elementary substructures (p45, 2002 edition) as : Suppose that M is a substructure of N. Then M is an ...
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111 views

Is there a “computable” countable model of ZFC?

Question Assuming ZFC is consistent (has a model), does there exist a set $S$ and a binary relation $\in_S$ on $S$ that satisfy the following? $S \subseteq \{0,1\}^*$ (this is the Kleene star, and ...
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Model-finding: negated quantifiers

I want to find a model and a countermodel for the following formula: $¬∀a¬∃b((P(a)∧P(f(b)))→Q(f(f(b))))$ I tried: Model 1: $A = \{x, y\}, P^M = \{x,y\}, Q^M = \{x\}, f(b) = b$ which satisfies the ...