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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Ordering of $\mathbb{R}$ not quantifier-free definable in $L_{R}$

I'm reading David Marker's book "Model Theory: An Introduction" and I'm trying to solve Exercise 3.4.24 which is stated as follows: Let $x$ and $y$ be algebraically independent over $\mathbb{R}$. a) ...
4
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2answers
41 views

Formula for perfect squares spectrum.

I have been working on exercises from "A first Course in Logic" by S. Hedman. Exercise 2.3 (d) asks to find a first-order sentence $\varphi$ having the set of perfect squares as a finite spectrum. But ...
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Models of $T$ in cardinallity $\kappa$ are isomorphic

Assume that $T$ is a consistent set in a countable language $\mathscr{L}$ with no finite models. There is a cardinal $\kappa$ such that every two models of $T$ with cardinallity of $\kappa$ are ...
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Are there number systems corresponding to higher cardinalities than the real numbers?

As most of you know, the set $\omega$ with cardinality $\aleph_0$ corresponds to what we normally know as the natural numbers $\mathbb{N}$, and the set $\mathcal{P}(\omega)$ with cardinality ...
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41 views

Is this theory complete?

I have a language $L=\{P\}$ with equation, where $P$ is binary predicate symbol. Language's formulas are: $\varphi \equiv \forall x \forall y (\neg P(x,x) \land (P(x,y) \to P(y,x)))$, $\psi \equiv ...
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2answers
70 views

Adding Substructures by Forcing

Consider a first order language $\mathcal{L}$ and an $\mathcal{L}$-structure $M$. Let $V$ be a model of ZFC (or ZF) the general question is that what would happen to classes, $Sub(M):=\{N~;~N~\text{is ...
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1answer
39 views

Equivalence Relation on a Class

I'm trying to understand Scott's proof of the incompatibility of axiom of constructibility and the existence of a measurable cardinal. I'm stuck in the use of Łoś's Theorem in the universe. Jech's ...
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1answer
73 views

Why are models in logic called models?

A model is an interpretation of a given formal language under which any wff in a given set of wffs of this formal language is true. Why are models called models? What's the reasoning behind the name? ...
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27 views

Is $F(x_1,x_2,\dots,x_n)$ where $(x_1,x_2,\dots,x_n)\in \Delta$,a semi-algebraic function?

Given $$F(t_1,t_2,\dots,t_n)=\int\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}dx_1dx_2\dots dx_n$$ where $P_1(x_1,x_2,\dots,x_n), P_2(x_1,x_2,\dots,x_n)$ are polynomials whose coefficients ...
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1answer
87 views

Uncountable reals in the theory

The Question I'm looking for a possibility to somehow proof the "essence" of Cantor's diagonal argument within a recursive first-order theory which is satisfied by the reals (better: within a theory ...
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33 views

A universal formula is not equivalent to an existential formula

Suppose a formula is looks like the following: $\forall x_1 ... \forall x_n \alpha$ Where $\alpha$ is a formula free of quantifiers. And if $P$ is a 1-ary relation letter, then the formula ...
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2answers
205 views

Can it be decidable for any polynomials to have the intersecting point?

Give system of polynomials$$P_1(x_1,x_2,\dots,x_n)=0,$$$$\vdots,$$$$P_k(x_1,x_2,\dots,x_n)=0$$ Can it be decidable for thoses polynomials to have the intersecting point ?
2
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1answer
36 views

Algebraic invariants for first order equivalence between fields

I know that every two models of the theory $ACF$ (namely two algebraic closed fields) with the same characteristic are elementary equivalent. But what about generic fields? Are there any algebraic ...
2
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27 views

Types linearly-ordered by deduction?

I'm wondering whether anyone has come across the following concept before: Consider a first-order language $L$ and a type $p$ over a theory $T$. I say that $p$ has a well-ordered filter-base if there ...
3
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33 views

Uncountably Categorical Theories and Embeddings

Suppose that $T$ is uncountably categorical. By the Baldwin-Lachlan Theorem, we note that $I(T,\aleph_0)=1$ or $\aleph_0$. Suppose that $I(T,\aleph_0)=\aleph_0$. Is it always the case that we get ...
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1answer
28 views

Use of model theory in flag algebras

I need to learn about Razborov's "flag algebras" (see http://bit.ly/1u1a1NB) to solve a problem about graphs. Flag algebras are a very general new algebraic tool for studying combinatorial structures. ...
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72 views

Is there a rationality-preserving order isomorphism between $\mathbb{Q}$ and two disjoint open intervals?

I have a homework question in a intro logic course, part of which requires me to Find an order preserving isomorphism between $\mathbb{Q}$ and $\mathbb{Q} \cap ((0,1) \cup (2,3))$. So, I need an ...
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1answer
41 views

Model-theoretic question about language of field theory.

Let $\mathscr{L}=\{+,·\}$ be the language of the theory of fields. Let $\phi$ be a sentence in this language. Show, using the compactness theorem of first-order logic, that if $\phi$ holds in finite ...
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1answer
38 views

Show that the function is an isomorphism between two $L$-structure.

The function: $$f: \mathbb{R} \longrightarrow (-1, 1)$$ $$ x \rightarrow \frac{x}{1 + |x|}$$ is an isomorphism between $\langle\mathbb{R}, <, =\rangle $ and $\langle(-1, 1), <, =\rangle$ where ...
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1answer
46 views

Logical Consequences and Ordered Fields.

How do I show that these two: $1.$ $\forall x(0 < x \rightarrow (-x) < 0)$ $2.$ $\forall x \forall y \forall z((x<y \wedge z<0) \rightarrow (y *z) <(x*z))$ are logical consequences ...
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1answer
23 views

Exercise 20.7 of Sacks's Saturated Model Theory (Partial isomorphisms)

I'm trying to solve the exercise in the title and I think it makes no sense. Here's what it says: An onto map $f: X \to Y$ is called an elementary partial isomorphism between $\mathcal{A}$ and ...
2
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56 views

Non-forking frames in AEC

Here http://shelah.logic.at/files/875.pdf on page 15, item 4 in the proof of 2.2.6, I would like to know why $S(M)\leq \lambda \times \lambda^+$. I understand that models in $K$ have cardinality ...
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34 views

An affine group behaving like a field

This question is about an example of interpreting a field in an affine group, from Section 1.3 of Marker's Model theory: An introduction. Let $F$ be an infinite field and $G$ be the group of ...
3
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1answer
58 views

Natural Algebraic Structures on the Set of Automorphisms of a Structure

If $M$ is a first order structure (e.g. some algebraic structure) we usually refer to its set of automorphisms, $Aut(M)$, as a group with its natural "function combination" operator.i.e. $\langle ...
2
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1answer
33 views

Model theory of valued

I am currently reading these notes on model theory of valued fields, in the section 3.3 appears this theorem: Theorem. Let $K$ and $L$ be valued fields, with residue fields $k_K$ and $k_L$ ...
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35 views

Relations between equations in a theory, and the number of independent equations

I have a question on equational reasoning in theories, which is made quite often in mathmeatics, and I am trying to make this more formal. So for my attempt to make this more rigouros, I choosed ...
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47 views

Books and articles on model theory for set theory

I'm interested in books and/or articles which explore a little more in depth the model theory of set theory. I'm aware that most books on set theory have a section or two on models (e.g. Jech, Kunen), ...
5
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46 views

Type-definable Forcing or forcing in a non-first order setting

Roughly speaking, in set forcing the forcing notion is a set from ground model's perspective and in class forcing its a definable subset of the ground model given by solutions of some formula with ...
1
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1answer
60 views

Reference request basic logic/model theory

I'm taking a knowledge representation class and need more perspective on basic model theory. We're currently using Levesque and Brachman. Specifically, a question on the midterm was something like, ...
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1answer
40 views

Cardinality of the set of non isomorphic structures of fixed cardinality

Let $L$, be a language and $\alpha$ be a cardinal; let $\Gamma:= \{\text{set of non isomorphic $L$ structures, having cardinality $\alpha$}\}$. Prove that $\operatorname{Card}(\Gamma)\leq ...
4
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2answers
46 views

Colored operads as finitely essentially algebraic theory.

I call a planar operad what is also called planar (multi-)coloured operad or multicategory and symmetric operad a symmetric multicategory or symmetric (multi-)colored operad. I have two questions ...
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1answer
32 views

Characterization of a theory whose model has elementary submodels as only its submodels

This is a problem (2.5.12) from Marker's Model Theory: An Introduction of showing that a model has only elementary submodels as its submodels if and only if for every formula is equivalent to some ...
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23 views

$n$-types of a structure

I got introduced to $n$-types of a structure a few weeks ago, but I can't really get my head around it. In an exercise I am asked the following: Define the binary relation $=_2$ on $\mathbb Z$ by: ...
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2answers
34 views

Proving the completeness of a theory $\Gamma$

Given a set of sentences $\Gamma$ in a first-order-language $\mathcal{L}$, such that for all structures $\mathcal{A}=(A,\ldots)$ and $\mathcal{B}=(B,\ldots)$, if both $\mathcal{A}$ and ${\cal B}$ ...
2
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1answer
31 views

Marker Exercise 2.5.10: universal part of a theory and supermodel

I'm trying to solve Exercise 2.5.10 in Marker's Model Theory: An Introduction. It goes: Let T be an $\mathcal L$-theory and $T_\forall$ be all of the universal sentences $\phi$ such that $T ...
0
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1answer
47 views

Any substructure of $(\mathbb{N}; 0, 1, +, \cdot)$ is itself

Consider a substructure $\mathcal{M} \subseteq \mathcal{N} = (\mathbb{N}; 0, 1, +, \cdot)$. Prove that $\mathcal{M} = \mathcal{N}$. EDIT: This result seems intuitively easy, but I'm having trouble ...
2
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1answer
48 views

An example of a formula with infinite Morley Rank

Given a Theory and a Model, you can define the Morley Rank of formulas with parameters from the model. I'd like you to give me an example of a formula (with theory and model) with infinite Morley ...
2
votes
1answer
54 views

A fragment of Exercise 1.3.4 in _Shorter Model Theory_ by Hodges

The following is what I believe is necessary to solve Exercise 1.3.4 in Shorter Model Theory by Hodges. Given two structure $\mathcal {A, B}$ of the same signature $\tau$, a set $S$ of generators of ...
5
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1answer
53 views

O-minimal Theories with Non-Dense Order Type

In this paper, Knight, Pillay, and Steinhorn prove that for any O-minimal structure $\mathfrak{A}$, in which the underlying order types is dense, and if $\mathfrak{B} \equiv \mathfrak{A}$, then ...
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36 views

Perron-Frobenius theorem for real closed fields via model theory

The Perron-Frobenius theorem states that any matrix over the reals with positive entries has at least one positive eigenvalue (and a bit more). The easiest proof that I know of runs as follows: any ...
18
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1answer
553 views

Murder at Hilbert's Hotel!

I'm sorry if this is a duplicate in any way. I doubt it's an original question. Due to my ignorance, it's difficult for me to search for appropriate things. Motivation. This question is inspired by ...
3
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2answers
70 views

Formal theories dealing with non-commutattive and non-transitive notion of equality

This question is inspired by a philosophical discussion which I don't want to bother you with. As far as I know when we use (or define) the statement "$x$ is equal to $y$" in logic and ordinary ...
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Can a model (of a general theory) be viewed as a (less general) theory?

Let me explain my question on an example. As a general axiomatic theory, consider group theory. A model for group theory is, for instance, group SO(3). But group SO(3) has its own axioms, so can we ...
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112 views

An impressive fact expressible in presburger arithmetic?

Is there something expressible in presburger arithmetic that would seem impressive to students at an undergraduate level?
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127 views

What are all kind of “metamath” good for? Can it help me here? [closed]

Those logical theories, which deals with questions that isn't really mathematics but reach mathematics more or less, often seems to be like textbooks full of definitions, plus some theorems of the ...
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1answer
63 views

Difference between type and similarity type

In usual terminology, is there a difference between the type and similarity type? Is there a general consensus for the definition of the two terms? Please suggest to me books where I can study these ...
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45 views

substructures generated by constant symbols

I am have recently started to learn about model theory, so this might be a stupid question. To learn model theory, I am reading David Marker's Model Theory. This is the situation in the proof of ...
1
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3answers
76 views

Is the (first order theory) of Hilbert spaces categorical?

Suppose the axioms of a Hilbert space (i.e. vector space, inner product, completeness and separability) are formulated as a first order theory. It can be shown that any infinite dimensional Hilbert ...
3
votes
1answer
50 views

Uncountable Dense Linear Orders

Is there an example of two uncountable equipollent dense linear orders without endpoints that don't satisfy the same first order properties? Or is it true that two uncountable equipollent dense linear ...
0
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1answer
53 views

If no interpretations satisfy a set of formulae U, is it possible for $U\models A$?

Note: '$ \models$' denotes logical consequence, defined as If $U \models A$, then $A$ is a logical consequence of $U$, if and only if every interpretation that satisfies U also satisfies $A$, ...