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 ...

learn more… | top users | synonyms

7
votes
1answer
155 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
223 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
votes
1answer
187 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
227 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
214 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
174 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
64 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
138 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
316 views

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

How many automorphisms the complex numbers field has?
6
votes
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 ...
2
votes
3answers
108 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
votes
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
votes
1answer
127 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 $\alpha$ ...
4
votes
1answer
90 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
93 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
146 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
267 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
114 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
89 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
152 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
71 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
34 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
105 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. ...
6
votes
2answers
255 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
193 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
464 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
157 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
126 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
44 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
77 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
117 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 ...
1
vote
1answer
122 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
175 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
114 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
109 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
230 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 ...
1
vote
0answers
121 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? ...
0
votes
1answer
39 views

modeltheory, union

There's a row of models, namely $M_0 \subseteq M_1 \subseteq M_2 \subseteq \dotsb$ and there's a theory $T$ such: $\phi \in T \rightarrow \phi = \forall x\exists y\,\psi$ with $\psi$ quantor-free. ...
9
votes
2answers
348 views

Can a model of set theory think it is well-founded and in fact not be?

ZF's axiom of regularity implies that no infinite descending sequence of sets $x_1 \ni x_2 \ni x_3 \ni \cdots$ exists. Precisely this theorem asserts the non-existence of a map from $\mathbb{N}$ to ...
1
vote
1answer
109 views

Renaming the elements of a mathematical structure

One of the most basic insights about mathematical structures is that we can rename their elements without fundamentally changing the structure. Question. How do we actually formalize this observation ...
1
vote
2answers
71 views

Logic: some basic plane geometry

Suppose you've got the language of some basic plane geometry, i.e. two 1-place relation symbols $P$ and $L$ for point and line and one 2-place relation symbol $I$ for point $x$ lies on line $y$. Now, ...
1
vote
1answer
176 views

Logic: $\text{Mod } \Sigma$, the class of all models of $\Sigma$ and $\text{Th Mod }\Sigma$, how do these relate?

While reviewing a question I had asked earlier here: Proving and understanding the Fixed point lemma (Diagonal Lemma) in Logic - used in proof of Godel's incompleteness theorem I have the ...
2
votes
1answer
404 views

Logic: cardinality of the set of formulas

How can you proof that $||L||=|L|$ if $L$ is infinite (where $||L||$ stands for the cardinality of the set of all $L$-formulas and $|L|$ the number of all constants, function and relation symbols)? ...
0
votes
3answers
76 views

Logic: defining 2 in a certain structure

Let $L= \{ \leq \}$ be the language of the partial orders and $M$ the $L$-structure with $M=\{ 1,2,3,4,6,12\}$ and $\leq_M=\{(x,y)$: $x$ is a divisor of $y$$\}$. Now, give an $L$-formula $\phi(x)$ ...
2
votes
2answers
126 views

Logic: building a sentence

Let $L$ be a language with a 1-place function symbol $f$. Give an $L$-sentence $\phi$ that is true in every $L$-structure $M$ if the following holds: if $M \models \phi$, then $M$ is infinite. My ...
2
votes
0answers
124 views

What kind of logics satisfy the coincidence lemma?

Lets formulate the incidence lemma as follows. We have a possibly infinite set of variables X and the domain of discourse U. Lets define an interpretation of the variables X in the domain U as a ...
5
votes
1answer
104 views

Definability in exponential fields assuming quasiminimality

I've watched a lecture by Alex Wilkie on the MSRI website which covered the first order theory of exponential fields. He gave an exercise which looks very surprising. It should be solvable, however, ...
6
votes
2answers
306 views

Two definitions of strong homomorphism

We say that $f: A \to B$ is a homomorphism iff it preserves the operations and relations of the structure: $f(o^A(\bar{a})) = o^B(f(\bar{a}))$ where $a ∈ A^{ar(o)}$, $o$ any function symbol from the ...
4
votes
1answer
182 views

Crashcourse Models in Set Theory

I am currently working through the lecture notes. In the end we had a short introduction to Models in Set Theory, but since it was quite to the end, we did not really go into details. So my ...