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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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
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122 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 ...
3
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157 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
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214 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 ...
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166 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
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134 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 ...
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261 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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126 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? ...
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41 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. ...
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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 ...
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1answer
114 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 ...
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73 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, ...
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1answer
182 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 ...
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1answer
536 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)? ...
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3answers
79 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
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2answers
133 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 ...
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156 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
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1answer
108 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
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434 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
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1answer
205 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 ...
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2answers
104 views

Do 'nice' first order logics have universal models?

A first-order logic is interpreted in a model where sentences of the logic can be said to be true or false. There may be more than one model, and we can identify morphisms between models. Do we have ...
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3answers
167 views

Can we define 'model' to mean 'diagram' in the sense of mathematical logic? If so, how to define the satisfaction relation?

According to Wikipedia, every $\sigma$-structure $A$ is associated with a collection of atomic and negated atomic sentences called its diagram. Here's the precise definition. For each σ-structure ...
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1answer
116 views

$dcl(A)=\{x \in M : x$ definable from $A \}$, Show $dcl(dcl(A)=A$

Marker exercise 1.4.10c $dcl(A)=\{x \in M : x$ definable from $A \}$, Show $dcl(dcl(A)=A$ I assume "$x$ definable from $A$" means "$\{x\}$ is $A$-definable." Definition 1.3.1: Let $\mathcal{M} = ...
7
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1answer
185 views

Non-standard proofs of standard theorems

In Richard Kaye's book Models of Peano arithmetic, one can read (page 13): We have proved that any nonstandard $M \models \mathrm{Th}(\mathbb{N})$ has a nonstandard $a \in M \models \theta(a)$ iff ...
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147 views

Running programs in nonstandard models of PA

I came across the following problem in several places, to paraphrase: Let $T$ be a recursively axiomatizable, consistent extension of PA. Then there exists some $e$ such that the $e'$th program ...
2
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1answer
83 views

Back and forth and the axiom of choice

Is the axiom of choice a necessary condition for the application of "back and forth construction" in model theory?
2
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1answer
74 views

Type $\sim$ Minimal Polynomial & Orbit

In Model Theory by Wilfrid Hodges, he gives an intuition of what a type is in the following way: "One can think of types as a common generalisation of two well-known mathematical notions: the ...
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75 views

Getting Failure of Compactness from the Failure of Upward Lowenheim-Skolem in Omega logic.

(1)Given Omega-completeness, and assuming compactness implies upward Lowenheim-Skolem, can one get the failure of compactness by showing failure of upward l.s.? I ask because it would seem to take ...
5
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1answer
302 views

Quantifiers as Adjoints in Generalized Logics

It is a well known fact that the classical universal and existential quantifiers can be seen as adjoints in certain categories. In the continuous model theory of metric structures (see ...
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55 views

Concurrent relation and enlargement

The superstructure $V({}^{\ast}X)$ [with respect to a monomophism $\ast : V(X) \to V({}^{\ast}X)$] is called an enlargement of $V(X) $ if for each set $A \in V(X)$ there is a set $B \in {}^{\ast} ...
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2answers
127 views

Theory vs. Deductive Theory

Looking through some notes on model and computability theory, I noticed that the definition of the term 'theory' changed between them; in particular, the model theory (based on Hodges' text) defined a ...
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92 views

Medium-strong (graph) homomorphisms

Weak (graph) homomorphisms are mappings $f: V(G) \rightarrow V(G')$ such that the images of connected nodes $x,y$ (in the source graph) are connected: $$R(x,y) \rightarrow R(f(x),f(y)) = R(x',y')$$ ...
4
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1answer
132 views

Do we assert the existence of set theory when reasoning about L-structures?

In model theory, if L is a first order language, by the definition of a L-structure $\mathcal{M}$ it is partly given by a non-empty set $M$ called the universe or domain of $\mathcal{M}$. From where ...
5
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1answer
373 views

Ehrenfeucht–Fraïssé game, how can I understand it?

My course of "Formal Methods" deals with Ehrenfeucht–Fraïssé games, particularly regarding the inexpressibility in FO logic. At the moment I've fully understand what this games are and how they are ...
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1answer
107 views

Question on the ultrapower construction of superstructure

On page 83-85 (here, on googlebooks), An Introduction to Nonstandard Real Analysis, Albert E. Hurd, Peter A. Loeb, two steps are given to construct a monomorphism $\ast : V(X) \to V({}^{\ast} X)$. The ...
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1answer
49 views

Signatures and L-Structures

Consider the field of real numbers $\mathbb{R}$. This is an $L$-structure. Is there such a thing as an $S$-structure (i.e. a signature structure)? Or because we can recover a first order language from ...
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221 views

Is there any countable ordinal number which has a member undefinable?

Let $Q\colon=\{\alpha \in Ord \mid \exists \beta \in \alpha(\beta \text{ is undefinable in } \langle\alpha,<\rangle )\}$ denote the set of all ordinal numbers which has an undefinable member. ...
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3answers
72 views

Show $s(s(a))=s(b)$ implies $s(a)=b$

Let us have a first order language $L=\{0,s\}$, where $0$ is a constant, $s$ is a function symbol of arity $1$. The first-order theory $T$ is axiomatized as follows: $\forall x \neg( s(x) = 0)$ ...
4
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1answer
372 views

A sentence that has infinite models, finite model, but no finite model above certain cardinality

Let $T$ be a theory and $\sigma$ a sentence, such that there exists infinite $\mathfrak{A} \models T + \sigma$. there exists finite $\mathfrak{A} \models T + \sigma$. there exists $n \in ...
3
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1answer
124 views

What's the idea behind the proof of saturation of internal sets via ultrapower construction?

I'm trying to understand the proof of saturation of internal sets via ultrapower construction on Robert Goldblatt's Lectures on the Hyperreals: An Introduction to Nonstandard Analysis. Though, it's ...
2
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1answer
66 views

Can I express (only) the syntactical formulation of a set theory within another.

Motivated by this question on the SE philosophy board "Is there a one true set theory", I want to ask for a clarification: I've seen e.g. ZF being expressed in Grothendieck set theory. If I say ...
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1answer
73 views

Every connected $\omega$-stable group has a zero element?

Let $G$ be a connected $\omega$-stable group and $p$ its unique generic. Let $a$ be a realization of $p$, $G\prec G_1$ an elementary extension containing $a$ and $q$ the non forking extension of $p$ ...
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1answer
85 views

Point evaluation of a linear functional on an Ultrapower

Let $E$ be a Banach space and $(E)_{\mathcal U}$ be an ultrapower for some ultrafilter $\mathcal U$ on an index set $I$. It is remarked in a paper that $(E')_{\mathcal U}$ can be naturally embedded ...
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525 views

Do isomorphic structures always satisfy the same second-order sentences?

I know that if two mathematical structures are isomorphic, then they satisfy the same first-order sentences. The converse is false. This is probably a completely obvious question, but is it true that ...
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1answer
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Is this a characterization of well-orders?

While grading some papers and thinking about a question related to well-orders (in particular, pointing a mistake in a solution), I came to think of a reasonable characterization for well-orders. I ...
6
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1answer
156 views

Model Theory (Hodges), Section 2.1, Exercise 13

Let $K$ be a field of characteristic 0, $n$ a positive integer, and $G$ the group $GL_n(K)$ of invertible linear transformations on $K^n$. Show that the following subsets of $G$ are ...
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1answer
135 views

Not Skolem's Paradox

Assume we have a countable, non-standard model of Peano Arithmetic (PA) in ZFC. http://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic Let $N^*$ be the universe of this model and let $m \in ...
4
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Crankery: Is there a perfect inner model of ZFC?

In the book "Aspects of Vagueness", the article "The alternative set theory and its approach to Cantor's set theory" by A. Sochor proposes the following definition: We will say that a set universe ...
5
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1answer
62 views

Restrictions of automorphisms to elementary substructures

Suppose that I have structures $M \preceq M'$ (in some first-order language). I have a set $A$, with $M \subseteq A \subseteq M'$, and an automorphism $f$ of $M'$. Is it is always possible to find ...
177
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1answer
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Is there a 0-1 law for the theory of groups?

For each first order sentence $\phi$ in the language of groups, define : $$p_N(\phi)=\frac{\text{number of nonisomorphic groups $G$ of order} \le N\text{ such that } \phi \text{ is valid in } ...