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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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 ...
3
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2answers
181 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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2answers
215 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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1answer
215 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
328 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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0answers
124 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
214 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 ...
3
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0answers
89 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
73 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
244 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
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2answers
134 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
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1answer
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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1answer
475 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
114 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
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3answers
296 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
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1answer
168 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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58 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 $L$?
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266 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
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1answer
98 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 ...
3
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2answers
149 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 ...
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2answers
232 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 ...
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1answer
166 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 ...
3
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1answer
121 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 ...
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149 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
91 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
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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
76 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
241 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
181 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
452 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
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1answer
236 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, ...
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0answers
179 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 ...
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1answer
87 views

Relations of language/theory/signature

Say that the language of the first order logic is the collection of symbols that can be used in the formulas + the grammar (the rules that specify how they can be combined)? 1) However, the signature ...
3
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1answer
180 views

Explanation of how models can differ on $\omega$?

Assuming set theory (here, ZF) is consistent, there is a model $V$ of ZF, the universe of all sets. So, there is a $\omega^V\in V$. A set $A\in V$ is countable iff a bijection $f\in V$ exists ...
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2answers
202 views

dense linear orders DLO

I am asked to prove that if I have two models of dense linear orders DLOs WITH the minimum and maximum. must be izomorpic to each other by fining direct izomorphy. I seem to always get stuck ...
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2answers
127 views

What's the name of this operator?

Let $f,g$ be functions in $C^A$ and $C^B$ respectively. Let $\boxtimes:C^A \times C^B \to (C\times C)^{A \times B}$ s.t. $f\boxtimes g(a,b)=(f(a),g(b))$ It seems not the tensor product, nor ...
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1answer
98 views

How to formally describe this Uppaal automata?

I have the following simple automata: What I'm looking for is a formal description of this based on the definition here $A=(\Sigma,\Gamma,S,s_0,\delta,\omega, F)$ How to declare all the ...
0
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1answer
563 views

Models of propositional logic

Define a theory of propositional calculus as the set $T$ of axioms (expressed in propositional calculus) and a set of valid symbols. What I would like to see are some examples of theories in ...
2
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2answers
115 views

Does Bernstein theorem hold for models with elementary functions?

Bernstein theorem is a general pattern that occurs in many areas of mathematics (see the Wikipedia article for some examples). Does it hold for arbitrary models with elementary embeddings? To be more ...
0
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1answer
81 views

Factor in ultraproduct

The general method for getting ultraproducts uses an index set I, a structure $M_i$ for each element i of I (all of the same signature), and an ultrafilter U on I. The usual choice is for I to be ...
2
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2answers
335 views

confusion regarding compactness theorem

I am getting somehow confused of compactness theorem. The compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This ...
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3answers
228 views

Proof that theory of infinite numbers does not have finite model

Just curious: Is there any proof that proves that there is no finite model of infinite numbers (Theory of infinite numbers)? Edit: by infinite number, I mean either Reals or Naturals.
2
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4answers
338 views

Model of theory of real closed field

I heard somewhere that models of theory of real closed field are isomorphic. However, there is also a statement in Internet which seems to say the opposite. Are the models of theory of reals ...
2
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1answer
235 views

Countability in first-order logic is relative to what exactly?

Skolem's Paradox tells us that countability in first-order logic is relative. Relative to what? Below is what I've gathered. Countability it relative to: 1. what a model takes to be $\mathbb N$ 2. ...
5
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4answers
1k views

Tarski's decidability proof on real closed field and Peano arithmetic

It seems very confusing that real closed field (which also can be used as the theory of real number) is decidable, while Peano arithmetic, which seems to be a subset of real closed field is ...
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1answer
129 views

Interpretation of nonlogical symbols in compactness arguments

Please don't give me a complete answer to the motivation part of the question. I want to figure that part out for myself. Motivation: As a starting example, say that a reversing function is a unary ...
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3answers
205 views

A quick question about categoricity in model theory

I just want to see if I am using the term "categoricity" correctly in the following context: (1) I was thinking about why someone might reject a simple resolution to Skolem's Paradox. (2) The ...
0
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1answer
94 views

Distinction between the universe of a model and the domain of a model?

Distinction between the universe of a model and the domain of a model? I'm pretty sure I'm wrong about this. But even reading Wiki, I'm still not clear. I'll use an example to illustrate what I ...
4
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1answer
130 views

expressiveness of computable infinitary logic

A language $L_{\omega_1\omega}$ generalizes an ordinary first-order language by allowing countably long disjunctions. If we take its nonlogical vocabulary to contain just a predicate for the ...