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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Model of a language $L$ vs. model of a theory $T$ in $L$

I am reading Just/Weese and they seem to use "model of a language $L$", for example, p. 90: and, more disturbingly, p. 91: Isn't this a "typo" (or perhaps sloppy writing)? If $L$ is any language ...
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51 views

Question about a proof that there is no formula $fin(x)$

Consider the following passage in Just/Weese: where $fin(x) \equiv $ "$x$ is a finite set". I'd like to confirm the following with you : requiring "ZFC $\vdash fin(n)$ if $n \in \omega$" is weaker ...
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70 views

Borel sets used in Model Theory

In Lemma 4.4.13 (page 161) of D.Marker- Model Theory book it is proved that $D(F,T)$ (the set of all possible F-diagrams of models of T) is a Borel set. Can someone explain to me the proof given a ...
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A theory $T$ is complete if and only if $Th(M) = T^\vdash$ for some model $M$

I'm trying to show that a theory $T$ is complete if and only if $Th(M) = T^\vdash$ for some model $M$ where $Th(M)$ denotes the set of all sentences that are true in $M$. What I have so far: ...
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Does $E$ in a model $\langle M, E\rangle$ of ZFC have to be wellfounded?

Consider the following exercise from Just/Weese: My first reaction was "Of course $E$ has to be strictly wellfounded otherwise it wouldn't model $\in$" but apparently I am missing something since I ...
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Confused about models of ZFC and passage of book

I have a question about a passage of Just/Weese. In the following, let $\mathcal M = \langle M, E \rangle$ be a model of ZFC. Here is the first half of what I'm about to ask a question: So I asked ...
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120 views

How to finish proof that $T$ has an infinite model?

I'm trying to prove the following: If $T$ is a first-order theory with the property that for every natural number $n$ there is a natural number $m>n$ such that $T$ has an $m$-element model then $T$ ...
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53 views

Equivalent characterisation of consistent

We call a set of formulas $\Sigma$ of a language $L$ consistent if there is no $\varphi$ in $L$ such that $\Sigma \vdash \varphi$ and $\Sigma \vdash \lnot \varphi$. Apparently, an equivalent ...
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Choosing a Master Thesis Topic: Logic - Model Theory

I am a first-year graduate student in maths. Around these days, I feel I must decide on which exact part of mathematics I shall go through. Infact, I have narrowed down the suitable options but still ...
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Complete first order theory with finite model is categorical

I am trying to prove that if $T$ is a complete first order theory that has a finite model then it has exactly one model up to isomorphism. To this end, I assumed that $T$ is complete with a finite ...
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237 views

The relation $<$ is not definable in the structure of the integers with successor function

I have to prove the following statement: Prove that there is no formula $\psi=\psi(x_0,x_1)$ in the language $\operatorname{Th}((\mathbb{Z},S))$ such that the relation ...
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242 views

Proof of compactness theorem

I wanted to prove the compactness theorem, p 79 Just/Weese: The (i) <= (ii) direction is not obvious to me. I thought I could prove it by showing not (i) implies not (ii) as follows: Assume ...
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322 views

Compactness Theorem Application

I am doing an exercise on the compactness theorem of first order logic. The task is to prove that there is no singe first order sentence which is satisfied in exactly the infinite graphs (thereby, a ...
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3answers
74 views

Understanding models and valuations

I'm thinking about part (a) of the following exercise in Just/Weese page 77: Here is the definition of valuation: For example, say we have a model of the language of group theory, $( \mathbb Z/ 2 ...
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324 views

Which sets are present in every model of ZF?

As in the title: the existence of which sets is implied by the axioms of $\mathsf{ZF}$? For example one such set would be the empty set whose existence is demanded by the Axiom of the Empty Set. But ...
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185 views

Property of finite structure inexpressibiliy in first-order logic

I want two show that a certain property $u$ of some finite structure is not definable in first-order logic. Is the following reasoning correct? Let $\mathcal{S}$ denote a finite structure. Further, ...
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128 views

Existence of elementary substructures of a uncountable structure over a countable language

Let $\kappa$ be an regular uncountable cardinal carrying a $\tau$-structure for some countable language $\tau$. What can be said regarding the existence of ordinals $\alpha <\kappa$ carrying ...
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72 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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99 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 ...
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161 views

Examples of vaughtian pairs

Greets I just want to see examples of vaughtian pairs. Thanks
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87 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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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 ...
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958 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, ...
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76 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 ...
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170 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 ...
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53 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 ...
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128 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 ...
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1answer
628 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. ...
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3answers
257 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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149 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 ...
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168 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 ...
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119 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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167 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 ...
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191 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?
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185 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 ...
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2answers
184 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 ...
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224 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 ...
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222 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 ...
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1answer
335 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 ...
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127 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 ...
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217 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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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
74 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 $F$:= $countable$ ...
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248 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 $\mathbb{N}$ ...
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135 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 ...
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113 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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57 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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489 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 ...
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115 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? ...
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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 ...