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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93 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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52 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 ...
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62 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, ...
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78 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 ...
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55 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
73 views

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
101 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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82 views

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
138 views

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 ...
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1answer
233 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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77 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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53 views

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 ...
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1answer
156 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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1answer
110 views

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
76 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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1answer
57 views

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
74 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
126 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
61 views

$\kappa$-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 ...
4
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1answer
263 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
54 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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3answers
98 views

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?
6
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1answer
166 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", ...
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2answers
108 views

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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1answer
598 views

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 ...
4
votes
1answer
107 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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votes
1answer
44 views

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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1answer
84 views

$\omega$-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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votes
4answers
177 views

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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0answers
40 views

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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votes
2answers
214 views

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 ...
2
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1answer
85 views

Number of models for some theory

Let $\mathcal L = \{ E(\_,\_) \}$ and $T$ be the $\mathcal L$-theory that says that $E$ is an equivalence relation with an infinite number of infinite classes. (I find this statement not clear, ...
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0answers
47 views

Given a Hardy Field is it always possible to find a smooth representative of each germ?

In this case I refer to a Hardy Field (of germs at infinity) $\mathcal{H}$ a a field of germs of real valued functions on $\mathbb{R}$ that is closed under differentiation. That is, if ...
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3answers
270 views

Showing that the Class of Cyclic Groups Aren't Axiomatizable

The class of finite cyclic groups are not axiomatizable, for if we supposed they were by some set of sentences $\Sigma$, then there would exist a model for $\Sigma$ of at least order $n$ for all $n ...
8
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1answer
160 views

What's the motivation behind saturated models?

In Model Theory by Chang & Keisler, saturated models are introduced on page 100. A model $\mathfrak U$ is said to be $\omega$-saturated iff for every finite set $Y \subset A$, every set of ...
4
votes
2answers
228 views

Why am I learning model theory?

This is kind of a big squooshy question (or series of questions), which I will try to cast in a more precise form. Apologies if I don't succeed. Context: I'm an amateur set theory/category theory ...
2
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1answer
198 views

Intuition behind isolated types

I learned that an $n$-type $\pi$ over a theory $T$ is isolated if there exists a formula $\varphi(x_1,\ldots,x_n)$ such that $T \cup \exists \bar x \varphi$ is coherent and $T\vdash \forall \bar x ...
2
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1answer
235 views

True and provably true sentences in a model. Are they the same thing?

In logic, it is said that each sentence in a (consistent) theory is either true or false in a given model. Checking the truth of a sentence in a finite model amounts essentially to finite enumeration ...
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2answers
228 views

Axiomatising Wellorder

It may be of use to recall what a strict total order is; namely a binary relation satisfying irreflexivity, transitivity and totality, as formalised below: $$\forall x(\neg P_{1}(x,x))$$ $$\forall x ...
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179 views

Defining “structured sets”

In his Notes on Set Theory (p. 44) Moschovakis defines: A structured set is a pair $U = (A,S)$ where $A$ is a set, the space of $U$, and $S$ is an arbitrary object, the frame of $U$. But even ...
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Countable Ultrahomogeneous Structures

I've been learning about countable ultrahomogeneous structures, where ultrahomogeneous means every isomorphism of finitely generated substructures extends to an automorphism of the whole structure. ...
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1answer
140 views

a consistent model of $\mathbb{N}$ that isn't?

(This question arose from a homework question which asked me to prove that (1st order) induction is independent from the other (1st order) Peano axioms) Let $\mathcal{L}$ be the language of Peano ...
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1answer
320 views

How many automorphisms $\mathbb{C}$ has?

How many automorphisms the complex numbers field has?
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1answer
112 views

Showing that a statement is absolute.

After reading about various properties of $V_\alpha$ and how it can be used to model various axioms of Set Theory, Kunen mentions that in $ZFC$, one cannot prove that there is an $\alpha$ such that ...
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3answers
112 views

Prove $\forall r \in \mathbb{R}. \exists k \in \mathbb{Z}. r < k$

I would like to prove that for every real number there exists an integer that is greater than it. My problem lies in that I am not sure how to construct the real numbers and provide their theory with ...
2
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2answers
93 views

Showing that the theory DTO is consistent

Toward the end of Kunen's Models of Set Theory section in his most recent Set Theory text, after talking about relativization, he begins to mention the idea of relative consistency proofs. I've been ...
3
votes
2answers
189 views

Using the Downward Lowenheim-Skolem-Tarski Theorem.

I've been reading about the models of Set Theory in Kunen's most recent Set Theory text, and working on exercises since this is my first time working with Model Theory. There is one exercise that I've ...
4
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1answer
128 views

A question dealing with conditions for which $V_\alpha$ models $ZFC$

I've been reading through models of Set Theory in Kunen's most recent Set Theory text and practicing exercises. He mentions that $V_\alpha$ can be used to satisfy certain axioms of $ZFC$ when $\alpha$ ...