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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66 views

Extend non-principal n-type

Is the following true? Let T be a complete theory in some elementary language. Let $n$ be a natural number and suppose $\Gamma$ is a non-principal $n$-type of T. Let $\Delta$ ne an $n+1$-type of T ...
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1answer
87 views

Prime Model and countable saturated model proof and types of a theory

I know that for the complete theory $Th((\mathbb{Q},+,0))$ we have the prime model $(\mathbb{Q},+,0)$ and the countable saturated model $(\mathbb{Q}^{\infty},+,0)$, but what should I do when I try to ...
1
vote
1answer
144 views

Examples of vaughtian pairs

Greets I just want to see examples of vaughtian pairs. Thanks
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2answers
82 views

Definability in FO($\mathbb{Q}, +, \leq$)

Is the set $\left\{(x, y) \in \mathbb{Q}_{>0}^2 \, : \, \frac{x}{y} \in 2\mathbb{N} - 1\right\}$ definable in FO($\mathbb{Q}, +, \leq$).
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3answers
227 views

Does the category framework permit new logics?

It appears to me that a topos permits a broader concept of subsets than the yes/no decission of a characteristic function in a set theory setting. Probably because the subobject classifier doesn't ...
9
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1answer
764 views

Show that binary < “less than” relation is not definable on set of Natural Numbers with successor function

After reading about the question, I've come to believe that it would suffice to exhibit and automorphism of that is not order preserving. However, I'm unsure of how to construct such an automorphism, ...
3
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1answer
75 views

Prime Model and boundary functions for n-types

Find a prime model and a countable $\omega$-saturated model of $Th((\mathbb{Q},+,0))$. Define a function from $\mathbb{N}$ to $\mathbb{N}$ such that, for each n, $Th((\mathbb{Q},<))$ has not more ...
2
votes
3answers
158 views

Number of types in a given complete Theory

Prove that $Th((\mathbb{Q},<,+,0,1))$ has uncountably many 1-types. Prove that $Th((\mathbb{Q},+,0,1))$ has countably many 1-types. Prove that $Th((\mathbb{Q},<,0,1))$ has five 1-types. Prove ...
2
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2answers
52 views

Definable orders

Let $(K, <)$ be an order field, can I define the order "<" in $K$ ? I know that $K \models 0<a \;$ if and only if there is $b$ in the real closure of $K$ such that $b*b = a$. Can I ...
2
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0answers
116 views

Realizing n-types of a complete theory

Let $\mathfrak{A}_0=(A_0,R_0)$ and $\mathfrak{A}_1=(A_1,R_1)$ be structures such that for each $i<2$, $A_i$ is a nonempty set and $R_i\subseteq A_i\times A_i$. We define the structure ...
3
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1answer
551 views

$2\mathbb Z$ is not a definable set in the structure $(\mathbb Z, 0, S,<)$

This is (a translation of) an excerpt from a model theory textbook that shows that $2\mathbb Z$ is not a definable set in the structure $(\mathbb Z, 0, S, <)$, where $S$ is the successor function. ...
2
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3answers
230 views

A theory with exactly $n$ countable models, for each $n>1$

For each $n>1$ we shall construct a first-order theory $T_n$ with exactly n countable models. Let $n>1$, consider the language $L_n=\left\{{R,c_1,...,c_n}\right\}$, where $R$ is a binary ...
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vote
1answer
138 views

Problem with Morley's Theorem

Greets. Morley's theorem states that a theory which is categorical for an uncountable cardinal is categorical in all uncountable cardinals. My problem with the theorem is that I haven't found a ...
1
vote
2answers
164 views

Finite Set of Models

This is only directed towards logicians, model theorists etc.. I am reading "Model Theory" by Keisler and Chang and have encountered the following question. Let $\Psi = \{M_1,...,M_n \}$ be a finite ...
3
votes
2answers
108 views

First order sentence true in $\mathbb{Q}$ but not in $\mathbb{R}$.

I have the following assignment question: Find a first order sentence that is true in $\langle\mathbb{Q},+,\cdot ,0,1\rangle$ but not in $\langle \mathbb{R},+,\cdot,0,1\rangle$. Most of what I ...
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2answers
157 views

Boolean algebra spectrum

The first-order spectrum of a theory, is the set of cardinalities of its finite models. Finite models of Boolean algebras are informally n-dimensional cubes, therefore boolean algebra spectrum is the ...
3
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2answers
174 views

Does Löwenheim-Skolem fail for $\mathcal{L}_{\omega_1\omega}$?

Does descendant or ascendant Löwenheim-Skolem fail for $\mathcal{L}_{\omega_1\omega}$ -logic?
2
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2answers
179 views

statement that is consistent in ZFC but the negation of it can be both consistent and inconsistent in ZFC or vice versa

Is there any known case where ZFC system is known to be consistent with a statement, but is also consistent with the negation of the statement? Or vice versa. Also, when we say ZFC is consistent with ...
2
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2answers
156 views

Existential quantification as projection

I read the following statement here regarding equivalence of existential quantification and projection of basic relations in model theory. The operation of taking image under a coordinate ...
4
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2answers
179 views

Gentle introduction into stability and classification theory

I am badly looking for a (very) gentle introduction into stability and classification theory answering at least some of the following questions: Why is a stable theory called "stable"? What is a ...
6
votes
1answer
196 views

Łoś's Theorem holds for positive sentences at reduced products in general?

Let $ \mathcal{L} $ be a language for first-order logic whose logical primitives are $ \neg$, $\vee$, $\wedge$, $\forall$, and $\exists$, with the usual formation rules. A sentence $ \sigma $ is ...
1
vote
1answer
261 views

Equivalence to a universal formula

I am trying to prove the equivalence of the following assertions (Exercise 2.5.12 from Marker "Model Theory: An Introduction"). There is a universal formula $\psi(\bar v)$ such that $T \models ...
4
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0answers
120 views

Some intuition behind o-minimal systems.

I am following the definitions of an o-minimal system as defined in "Tame Topology and O-minimal Structures" by Lou van den Dries. It is immediate from the definition that the graph of $\sin(x)$ is ...
10
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1answer
204 views

Do ordered fields and archimedian ordered fields have the same first-order theory?

Let us suppose that the first-order language of ordered fields has symbols for addition, subtraction, multiplication and order, and constant symbols for 0 and 1. An ordered field is said to be ...
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0answers
82 views

An application of Descriptive set theory in Model theory.

In page 162 of D.Marker Model theory book he proved that the set $S_n(F,T)$ of all $F$-types realized by some $n$-tuple in some countable model of $T$ is analytic (this is with any $F$:= $countable$ ...
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1answer
72 views

Question about a defined function in D.Marker Model Theory book.

In page 162 of D.Marker Model theory book he proved that the set $S_n(F,T)$ of all $F$-types realized by some $n$-tuple in some countable model of $T$ is analytic (this is with any $F$:= $countable$ ...
3
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1answer
230 views

How to exhibit models of set theory

Even though $\mathbb{N}$ cannot be defined by first order means, it can be defined by second order means. Anyway: it can be defined, and there is no doubt, which abstract structure $\mathbb{N}$ ...
5
votes
2answers
133 views

Nonstandard extension of a function with a limit

Question 1. Let $g : \mathbb{R} \to \mathbb{C}$ with $g(y) = \lim_{x \to \infty} f(x,y)$, where $f : \mathbb{R}^2 \to \mathbb{C}$. Is it correct that the nonstandard extension $^*g$ will have $x \in ...
2
votes
1answer
103 views

On the Decision Problem for Two-variable First-Order Logic

I have a question concerning the model construction of the $\forall \forall \land \forall \exists$ - Scott sentence on page 6 in this paper: www.cs.rice.edu/~vardi/papers/basl96.ps.gz Why do we ...
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votes
0answers
54 views

Morphisms of Euclidean Geometry

In Euclidean geometry $\mathbb{E}^2(\cong \mathbb{C})$, the group $G=\{z\mapsto raz+b, z\mapsto ra\bar{z}+b\colon a,b\in \mathbb{C},|a|=1, r\in \mathbb{R}^{+} \}$ is precisely the group of ...
5
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1answer
414 views

Is there a difference between a model and a representation?

I'm thinking of models in logic here, vs. e.g. group representations. Is there a difference between a model and a representation? Could one not explain both at the same time? A model gives an ...
4
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0answers
111 views

is a group that eliminates quantifiers in the group language Abelian?

Let $G$ be a group. Consider it as a structure in the language of groups $L=(e,\cdot, (-)^{-1})$. Suppose that $G$ eliminates quantifiers in this language. Is it true that $G$ must be Abelian then? ...
5
votes
3answers
277 views

How are the full semantics of SOL and HOL specified?

In relation to this question about the "fundamental" character of possible logical systems, I realized that I just had an intuitive (and so, inadequate) understanding of the way logics higher than FOL ...
3
votes
1answer
159 views

question about Skolem theories

Right now I am reading a proof of Downward Löwenheim-Skolem theorem in Hodges, but I am slightly confused about a proof Hodges makes. Let me write down some of the definitions. Definition: Let ...
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0answers
53 views

Density for archimidean extension of real closed field

Let $(k,<)$ be a real closed field and $L|K$ an ordered extension such that $\forall x\in L \exists y\in k\; (x<y)$. Is $k$ dense in $L$?
2
votes
3answers
251 views

Weak categoricity in first order logic

In a certain sense, only finite structures are definable up to isomorphism in first order logic. But if we rely on a metatheory containing a sufficient strong set theory (like required for second ...
4
votes
1answer
97 views

the solution of ode can be encoded as first order logic

Consider the following sytem of ODEs $\dot{x}= Ax$, and given $x(0)$, where $A$ is a $n\times n$ matrix with rational entries. Can I encode the solution, say $x(t)$ for a given $t$, as a first ...
2
votes
2answers
142 views

Am I allowed to realize one object twice within one set-theory?

Say I consider a set theory with the Axioms of Extensionality and the Axiom of Pairing. As I understand it, stating the axiom allows me to make a definition like $$(a,b):=\{\{a\},\{a,b\}\}$$ and ...
3
votes
2answers
223 views

Why is completeness theorem true?

Who can teach me completeness theorem? Thanks! Recommending a book is also welcome. More specifically, it says that if a statement is true in all models of a theory, then it has a proof from this ...
2
votes
1answer
154 views

first order logic question model

suppose we have a model for a language in first order logic $ M=<D,I> $ such that D is the domain and I is the interpetation such that for every $ a \in D $ we have a closed noun (a noun with no ...
2
votes
1answer
112 views

Absoluteness and categories

From the wikipedia article on the Skolem paradox: A central goal of early research into set theory was to find a first order axiomatisation for set theory which was categorical, meaning that the ...
10
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1answer
143 views

Saturated Boolean algebras in terms of model theory and in terms of partitions

Let $\kappa$ be an infinite cardinal. A Boolean algebra $\mathbb{B}$ is said to be $\kappa$-saturated if there is no partition (i.e., collection of elements of $\mathbb{B}$ whose pairwise meet is $0$ ...
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2answers
84 views

Logic question proving something about compactness

Let $\Sigma$ be a set of formulas. There's a finite set $\Lambda \subseteq \Sigma$. I'm asked to prove or disprove that $\Sigma$ has a model if and only if $\Lambda$ has a model. It seems to me ...
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9answers
982 views

Learning Model Theory

What books/notes should one read to learn model theory? As I do not have much background in logic it would be ideal if such a reference does not assume much background in logic. Also, as I am ...
2
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2answers
74 views

Embedding of standard model of arithmetic to PA-model

I am working on the following problem: Let $ S_{Arithmetic} = \{+, *, 0, 1\}, \mathfrak{M} $ a model for PA (first-order peano axioms) }, and $ \mathbb{N} = (\mathbb{N},+ ^{\mathbb{N}}, ...
4
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1answer
222 views

Lindenbaum algebra is a free algebra

The following is a continuation of this question. I would like to prove that the Lindenbaum algebra is a free algebra. Hopefully I would like to hear hints on how to proceed in the 'right' ...
4
votes
1answer
165 views

Are there simple counterexamples to a strengthening of omitting types theorem

The famous Ehrenfeucht's omitting types theorem states that for any countable set of nonisolated types (without parameters), there is a (countable) model such that it does not realize any of them. A ...
3
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1answer
372 views

Complete/incomplete theory

I am thinking about completeness and incompleteness of theory's, and to illustrate both properties i am thinking of how to build an complete system, and then turn it into an incomplete one. Example. ...
2
votes
1answer
203 views

Is there a relationship between the Compactness Theorem and the upward Lowenheim-Skolem Theorem in FOL?

Is there a relationship between the Compactness Theorem and the upward Lowenheim-Skolem Theorem in FOL? I was thinking of another post of mine "Why accept the axiom of infinity?" when I though, ...
2
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0answers
160 views

Can we find a nice definition of Congruence in Topology?

According to my knowledge, quotient structure is a original structure divided by a congruence. However, quotient topology space is defined this way. Quotient_topology In this way, $\sim$ is only said ...