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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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
100 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 ...
6
votes
1answer
510 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
103 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 ...
0
votes
1answer
43 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 ...
1
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1answer
69 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 ...
5
votes
4answers
158 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 ...
2
votes
0answers
38 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 ...
4
votes
1answer
162 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
votes
1answer
81 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, ...
2
votes
0answers
40 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 ...
5
votes
3answers
211 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 ...
7
votes
1answer
126 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
198 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 ...
0
votes
1answer
119 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
votes
1answer
199 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 ...
4
votes
2answers
184 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 ...
3
votes
0answers
148 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 ...
2
votes
0answers
55 views

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. ...
2
votes
1answer
128 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 ...
3
votes
1answer
293 views

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

How many automorphisms the complex numbers field has?
7
votes
1answer
107 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 ...
2
votes
3answers
107 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 ...
3
votes
2answers
92 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 ...
4
votes
2answers
182 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 ...
5
votes
1answer
121 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 $\alpha$ ...
5
votes
1answer
79 views

Finding a counterexample in model theory

I'm currently reading about models of Set Theory, and I'm working on exercises to better understand the concepts. In Kunen's most recent Set Theory text, he mentions that if we have a transitive model ...
1
vote
1answer
80 views

“Amalgamation” of elementary equivalent structures

I've been asked this as an "easy exercise" on diagrams. Let $\mathfrak M_1, \mathfrak M_2$ be two structures over the same language $\mathcal L$. Prove that $\mathfrak M_1 \equiv \mathfrak M_2$ if, ...
4
votes
2answers
129 views

Find a set of sentences $\Sigma$ such that the set of all models of $\Sigma$ is countably infinite.

I'm just starting Model Theory (Chang & Keisler) and I'm having trouble right off the bat with exercise 1.2.9 (ii): Give an example of a set $\Sigma$ of sentences such that the set of all ...
0
votes
1answer
211 views

Showing the every consistent set of sentences has a model

I want to do a short proof showing that every consistent set of sentences has a model. I am assuming the derivability version of completeness for first-order logic, in for form: $$\Sigma \models F ...
3
votes
2answers
113 views

Does the language of group theory need a constant?

I would like to verify the understanding of my claim. In Model Theory: An Introduction by David Marker Example 1.2.5 defines the language of the theory of groups as follows to be $\mathcal{L}=\{., ...
3
votes
1answer
85 views

Is there a generic definition of “strongly indistinguishable”?

This is related to a previous question. Consider the quasiordered set $Q = \{\bot, q,q', \top\}$ with $q \lesssim q'$ and $q \lesssim q',$ such that $\bot$ is the unique least element and $\top$ the ...
0
votes
1answer
151 views

Quantification and logical relations, shorthand notation $\forall/\exists x \in M…$

I know the following shorthand: \begin{align*} \exists x \in M : P(x) & := \exists x ( x \in M \to P(x) ) \\ \forall x \in M : P(x) & := \forall x ( x \in M \to P(x) ). \end{align*} Now for ...
3
votes
0answers
67 views

Question on a Theorem from Chang-Keisler's Model Theory concerning $\Sigma^0_n$ sentences

The Theorem is 3.1.11 and states that for $n>0$ the following are equivalent : $\phi$ is equivalent both to a $\Sigma^0_{n+1}$ and a $\Pi^0_{n+1}$ sentence. $\phi$ is equivalent to a Boolean ...
1
vote
0answers
32 views

Graphs: First Order Characterisation Of A path

Whilst reading this: http://dtai.cs.kuleuven.be/krr/files/seminars/IntroToFMT-janvdbussche.pdf a seminar on finite model theory, I thought that something was wrong. "Given a Graph G and a Binary ...
2
votes
2answers
98 views

About a sentence in logic theory that I don't understand.

Can somebody explain to me the following terms in logic? I have to read a paper in combinatorics that says this, but I don't understand anything in this sentence, where the author speaks about logic. ...
5
votes
2answers
185 views

Is there a Second-Order Axiomatization of ZF(C) which is categorical?

A theory $T$ is called categorical if it only has one model upto isomorphism. (Note: this has nothing to do with category theory.) The Lowenheim-Skolem theorem states that no first-order theory with ...
3
votes
1answer
172 views

Models of real numbers combined with Peano axioms

Suppose you take the axioms for a Dedekind-complete ordered field and weaken the Dedekind-completeness axiom to the corresponding weaker first-order axiom schema (e.g. replace the left and right sets ...
1
vote
1answer
336 views

Neither Even Nor Odd Natural Numbers

I confused myself and the OP when I tried to answer a recent question. Modular arithmetic (MA) has the same axioms as first order PA except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)$. ...
6
votes
1answer
150 views

Elementary equivalence of polynomial rings

In his notes on the model theory of valued fields, Lou van den Dries mentions in bypassing that the polynomial ring over the complex numbers $\mathbb{C}$ is not elementarily equivalent to the ...
3
votes
1answer
122 views

An exercise in stability theory

This is taken from Pillay's highly minimalistic book on stability theory. Let $T$ be stable, $\mathcal{M} \prec \mathcal{N}$ models, and $a$ a tuple in the big model such that its type over $N$ ...
2
votes
0answers
43 views

What are some ways of showing that a structure is not minimal?

The question is really in the title. My background in model theory is very limited. Basically nothing past the definition of minimal structures and minimal subsets. I am interested in some ...
-1
votes
1answer
73 views

Elementry question on elementarily equivalence

Source: SHAWN HEDMAN Definition:Let M and N be V-structures. If M and N models the same V-sentences, then M and N are said to be elementarily equivalent, denoted $M \equiv N.$ Example: the ...
6
votes
1answer
115 views

Can there be an abelian group $G$ where $\bigcap_{n\in\mathbb N} nG$ is not divisible?

I heard a talk yesterday on MacIntyre's theorems, which involved a decomposition of ($\omega$-stable) groups into a divisible part and a bounded exponent part. Apparently there is a result of ...
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0answers
97 views

Theories with countably many countable models

Having another question in mind (which is not yet fully worked out, but will come soon) I'd like to gather some examples of (interesting) theories with countably many countable models ...
3
votes
1answer
156 views

Number of isomorphism classes of countable models of a theory

Whether there are countably or uncountably many isomorphism classes of countable models of a given theory depends on the theory: if the theory is strong enough, there will be only countably many ...
2
votes
1answer
98 views

What is a Proper Signature for a Vector Space?

The definition of a signature I'm working with is a quadruple $\sigma = (C,F,R,\sigma')$ with $C$ serving as a set of constant symbols, $F$ serving as a set of function symbols, and $R$ serving as a ...
4
votes
1answer
99 views

Why doesn't the definition of (model-theoretic) conservative extension need strengthening?

In the wikipedia page dedicated to conservative extensions, we find the following sentence: $T_2$ is a model-theoretic conservative extension of $T_1$ if every model of $T_1$ can be expanded to ...
6
votes
0answers
208 views

Characterization of the First Order Theory of Ordered Abelian Groups via Quantifier Elimination

In the paper "Elimination of Quantifiers in Algebraic Structures" Macintyre, McKenna and van den Dries, proved that every field (ordered field) whose theory admits quantifier elimination in the ...
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0answers
115 views

Question on Model completeness in FOL

Claim: Suppose T is model complete and has a model embeddable in every model of T. Show T is complete. This is in Sacks' book Saturated Model theory, problem 8.4. Is the following proof correct? ...