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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Set of formulas in Model Theory

I'm reading the book Model Theory by Chang and Keisler and there is one thing that always bugs me. Very frequently we have something like $\Sigma(x)$ representing the set of all formulas in a language ...
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85 views

Model theoretic answer for having algebraic closure

I am beginner at the model theory and I learn compactness theorem at the class and I saw some application of it and one of them is that "every field has an algebraic closure". How can I prove it with ...
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Recursive non-standard models?

Any algebraically closed field (ACF) is a model of Modular Arithmetic (MA). MA has the same axioms as first order Peano arithmetic except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)$. MA ...
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2answers
114 views

Compactness Theorem explanation

Compactness Theorem definition: If $T$ is a theory in a first-order language $L$, then $T$ has a model iff every finite subset $S$ of $T$ has a model. A number of questions regarding to this ...
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Why do we need sometimes other structures than mentioned in the theorem to prove theorems?

For example when one proves that the elementary theory of finite fields is decidable, one uses pseudo-finite fields which are not in generally finite fields. Why do we need such a larger fields to ...
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2-type not-realised in Q

my question is the following: given the additive group of rational numbers, i.e. $Q = \langle {\mathbb Q},+,0\rangle$ and $T$ the theory of $Q$, how can I find (explicitly) a 2-type which is not ...
3
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1answer
70 views

Finitely many countable models implies decidability

Suppose $T$ is decidably axiomatizable first order theory and has no finite model. We shall focus on countable models. If $T$ has just one countable model (up to isomorphism), which means $T$ is ...
2
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1answer
72 views

Can axioms of the Euclidean space be proven in the Real space?

I'm not a mathematical logic student so my question can be naïve. I'm thinking if we identify $\Bbb{E}^2$ with $\Bbb{R}^2$( as a normed space with the Euclidean norm), Then can we prove all Euclidean ...
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A Question Regarding Uncountable Standard Models of ZFC Where CH is False

Let M be an uncountable standard model of ZFC, let $\frak c$ be the cardinality of the continuum, and let (just for the sake of argument) $\mathfrak c=\aleph_2$. If one assumes M has 'all' the ...
2
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1answer
78 views

Showing a Theory $T$ is Substructure Complete

Let $T$ be a (complete and consistent) theory. Suppose $T$ exhibits the following two properties: (1) model-completeness: if $\mathcal{M} \models T$ and $\mathcal{A} \subseteq \mathcal{M}$ s.t. ...
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46 views

Introduction to Valued Fields

I'm looking for an introductory text on valued fields, to be used as the basis for a reading group for model theorists. Currently, I know of one such text, Valued Fields by Engler/Prestel. However, ...
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Showing that $(\mathcal{M}, N) \equiv (\mathcal{N}, N)$

Let $T$ be a theory (complete and consistent) and let $\mathcal{M} \models T$. Let $\mathcal{N} \subseteq \mathcal{M}$. Suppose we have that $\exists \mathcal{M}' \supseteq \mathcal{M}$ s.t. ...
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3answers
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Why can any type be realized?

I couldn't find this question asked previously, which means it's probably an especially daft question. Given an $\mathcal{L}$-structure $\mathcal{M}$, my textbook defines an $n$-type over $A\subseteq ...
3
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2answers
124 views

Should axioms be viewed as part of the signature?

I included category theory in the tags in order to get feedback from the categorial logic community. It goes without saying that this isn't really category theory. A semigroup can be defined as a ...
0
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2answers
134 views

Is there a useful application of Peano arithmetic?

If there is, can someone provide an example of how Peano arithmetic can be used to solve a real-world problem? If not, can someone provide an example of any axiomatic system other than ZFC that can ...
2
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1answer
52 views

Consistency first-order theories

is it true that any consistent first-order theory has a model? In case of affirmative response: 1) Is it the Godelian completeness proof? 2) Is there a standard strategy for constructing ...
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1answer
66 views

Ultraproducts by countably complete ultrafilter

I've recently learned about ultraproducts, but the source I learned from almost immediately after the definitions restricted to talking about countably incomplete ultrafilters. I know that the ...
4
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1answer
47 views

Which sentences survive the passage from $X$ to the set of all functions $I \rightarrow X$?

Suppose $X$ is a mathematical structure with a single underlying set which we will also denote $X$, equipped with some functions and relations. Letting $I$ denote an arbitrary non-empty set, we see ...
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Question about models theory

Which of the models M_1,M_2,M_3 in a picture is atomic? Which is saturated? For the two models that are not saturated, find 1-types which are omitted. For the two models that are not atomic, find ...
2
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1answer
54 views

Countably many worlds and Universal Sentence.

This is a naive, kind of informal argument. Suppose we have a language with just one predicate $P$ and constants $a_{1}, a_{2}, a_{3}$ and so on. Suppose also that we have countably many worlds $1, ...
2
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1answer
76 views

Is it true that $\Bbb Z^*\setminus\Bbb Z$ has no finite elements?

If we consider the hyperreals, we know that there exist non-zero infinitesimals so $\mathbb R^*\setminus\mathbb R$ has finite elements. However, it seems like that is not true for $\mathbb ...
2
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1answer
48 views

Comparing models through partial isomorphisms

Let $T$ be a theory of a language $\mathcal{L}$ with no function symbols. Let $\mathfrak{A}, \mathfrak{B} \models T$. For all finite sets $X \subseteq A$ and $Y \subseteq B$, there exists a function ...
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2answers
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extension of group operation from $\mathbb{Q}$ to $\mathbb{R}$

I'm having a hard time with this (seems easy, but could be misleading) problem: Let $A \subseteq \mathbb{Q}$ be a convex subset, and let $+$ group operation on $A$. Let $\overline{A} := \{x \in ...
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2answers
86 views

Equivalence between two definitions of infinitary logic

The common definition of $ \omega $-logic (a.k.a $\mathcal{L}_{\omega_1,\omega}$ logic) is the usual first order logic allowing infinite conjunctions and infinite proof. Chang and Keisler, in section ...
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Models of the full theory of a structure

I'm reading Model theory: an introduction, by David Marker. I'm at page 14, where it says: ...one way to get a theory is to take $\operatorname{Th}(\mathcal{M})$, the full theory of an ...
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2answers
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For two theories $T,T'$, what does $T\vdash Con(T')$ really tell us about the models of $T$?

Inspired by this question, which I realized I couldn't answer (because model theory and me don't get along). I've made a few edits to (hopefully constructively) tighten the question a bit. If for ...
6
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1answer
183 views

Using the Reflection Theorem

I've been reading about the Reflection Theorems in Kunen's 2011 Set Theory book. The idea that $ZFC \not \vdash \exists \gamma [V_\gamma \models ZFC]$, but $ZFC \vdash \exists \gamma [V_\gamma \models ...
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1answer
51 views

Omitting types theorem for types with parameters

Does the omitting types theorem as exposed e.g. in Hodges consider types with parameters or is it just about types over the empty set?
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Showing that $|\phi(\mathcal{N})| = \kappa$ s.t. $\mathcal{M} \equiv \mathcal{N}$ with $|\mathcal{N}| = \kappa$

Problem: Suppose $\mathcal{M}$ is an $L$-structure and $\phi \in L_n$ ($n > 0$) is such that $\phi(\mathcal{M})$ is infinite. Then show that for every cardinal $\kappa$ with $\kappa \ge |L|$ there ...
3
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1answer
129 views

Initial Segments of Modular Arithmetic

Modular arithmetic (MA) has the same axioms as first order Peano arithmetic (PA) except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)$. MA is $\omega$-inconsistent and all infinite models ...
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75 views

show $\equiv$ implies $\cong$ [closed]

Prove that for $L$-relational an arbitrary language, if $M$ is a finite $L$-structure and $M \equiv N$, then $M \cong N$. Do it for $L$ finite and then generalize it to infinite case. I am not able ...
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Showing that $\mathcal{M} \preccurlyeq \mathcal{N} \implies \mathcal{M} \equiv \mathcal{N}$.

Suppose that $\mathcal{M} \preccurlyeq \mathcal{N}$. Then by definition we have that $\mathcal{M}$ is a substructure of $\mathcal{N}$ s.t. for any (possibly empty) tuple $\overline{a}$ from $M^n$ and ...
2
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1answer
54 views

Can a sentence in a model-theoretic conservative extension be translated in the language of its reduct?

Let $L1$ and $L2$ two languages with $L1 \subset L2$ and $T1$ and $T2$ respectively a theory in $L1$ and $L2$. We say that $T2$ is a model-theoretic conservative extension of $T2$ iff every model $M1$ ...
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Showing that $|\phi(\mathcal{M})| = |\phi(\mathcal{N})|$ if $\phi(\mathcal{M}) \Subset M^n$ and $\mathcal{M} \equiv \mathcal{N}$

Let $\phi \in L$ define a finite set $X$ in the $L$-structure $\mathcal{M}$. Show that in every $\mathcal{N}$ elementarily equivalent to $\mathcal{M}$, the set defined by $\phi$ has the same power as ...
3
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1answer
58 views

Question about polynomials of odd degree with no zeros in formally real fields which are maximal to the property of being ordered

I have encountered this argument while reading Tent and Ziegler's "Course in model theory", and I don't know why it is justified. It arises during the proof that every ordered field has a real ...
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1answer
72 views

On deductively closed theories

Definition A theory $T$ is a set of sentences. A structure $\mathcal{A}$ is a model of $T$ if $\mathcal{A}\vDash T$. To better understand the situation, let us recall the classical Galois connection ...
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1answer
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$\kappa$-stable theories and number of types

How can I show that if $T$ is a $\kappa$-stable theory, then in each model of $T$, over every set of parameters with at most $\kappa$ many elements, there are at most $\kappa$ many n–types.
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Name of, and (if I'm lucky) references on, a particular property of an interpretation

So here I am studying the Ackerman interpretation (via Kaye-Wong) to try and suss what the fragment of arithmetic associated with KF (Mac Lane minus Foundation and Infinity, with separation restricted ...
2
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128 views

How can be a set of partial isomorphisms defined from a n-back-and-forth system?

While studying partial Ebbinghaus-Flum's Mathematical Logic, I came across the partial isomorphism definition, as build upon an $n$-back-and-forth system. Consequently, the question I raise in the ...
3
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1answer
48 views

Extended reals from ultraproduct of algebraic numbers

Let $\mathbb{A}$ denote the field of real algebraic numbers. Let $\mathcal U$ denote a free ultrafilter. Construct $F=\prod_{\mathcal U} \mathbb{A}$. This is a field containing $\mathbb A$, and we ...
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Is there a sentence in the language $\{xRy\}$ with only infinite models?

Can you find a sentence in a language with only a binary relation $R$, all of whose models are infinite?
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1answer
134 views

Question on existential sentences

A sentence is called existential if it is of the form $\exists x_1 \ldots \exists x_n \ \phi(x_1, \ldots, x_n)$, where $\phi$ is quantifier free. We know that (see Chang-Keisler "Model Theory", ...
5
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2answers
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define the reals in a non-archimedean elementary extension of the real field.

Can it be done? We have the real field $(\Bbb R,+,-,\times,0,1,<)$, of course $(0,1,-,<)$ are definable using the rest. We take an elementary non-archimedean extension. Can we define the ...
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What's the best way to teach oneself both Category Theory & Model Theory?

I've done a bit of reading around both Category Theory & Model Theory (CT & MT) as a novice in each field. I'm interested in how they might combine, particularly when applied to Algebra. [So ...
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99 views

Infinite set of standard primes as the set of standard prime divisors of a nonstandard number

Suppose $(N, +, \cdot, 0, 1, <, =)$ is a proper elementary substructure of $(N^*, +^*, \cdot^*, 0^*, 1^*, =^*, <^*)$. Show that there exists some (infinite) $b$, where $b ∈ N^*$, such that for ...
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What is $R(\omega)$ (and where can I find definitions for similar common notation)?

Model Theory by Chang and Keisler references $R(\omega)$ frequently, usually in the context of models $\langle R(\omega), \in\rangle$ of ZF. What does this notation mean, specifically? From the ...
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$\omega$-categoricity of a theory

Let $T$ be a complete $\omega$-categorical theory which have infinite models, and $C$ a $\omega$-saturated model of $T$. Let $A\subseteq C$ and $T_A$ be the theory of the model $C_A$, the structure ...
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4answers
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Is every theorem of PA true in the standard model of number theory $N$?

My understanding is that every theorem $\phi$ of $PA$ is true in $N$ because $N$ is a model for $PA$, $N\models PA$. By completeness of first order logic, "$PA\vdash\phi$" implies that "if $N\models ...
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Ways to build models with specific properties

I'm studying Model Theory: an introduction by David Marker and more specifically doing the exercises of chapter 2 ("Basic techniques"). In several of these exercises, one is asked to build new models ...
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First order theory of abelian groups and first order theory of cyclic groups are coincide?

Let $T$ be a first-order theory of cyclic groups. Even if an abelian group $(G,+)$ satisfy $(G,+)\models T$ there is no reason that $(G,+)$ is a cyclic. (For example, by Löwenheim–Skolem theorem there ...