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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Morley rank (with an unusual definition)

For a definable set $X \subseteq \mathbb{U}^n$, let us denote $\text{RM}(X)$ the Morley Rank $\text{RM}(\varphi(\bar{x}))$, with $\varphi(\bar{x})$ the formula defining $X$. Show that, for $X ...
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Does an interpretation of a structure by itself induce a bijection on the automorphism group of the structure?

Let $\Gamma$ be a model-theoretic interpretation of a structure $B$ in a structure $A$. Then $\Gamma$ induces a group homomorphism $\alpha_\Gamma:\mathrm{Aut}(A) \rightarrow \mathrm{Aut}(B)$. (See, ...
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Prove the Robinson arithmetic has infinite non-isomorphic models

I found this question: Can finite theory have only infinite models?, where is proved that Robinson's Arithmetic can have infinite models, but I've been unable to prove or find a proof of the existence ...
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Consequence in $\mathcal{L}_{\infty\lambda}$

Consider the infinitary first-order language $\mathcal{L}_{\infty\lambda^+}$ whose non-logical vocabulary consists of $\lambda \geq \omega$ individual constants and countably many predicate constants ...
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How would one prove that satisfaction of closed formulas is valuation-independent? (In FOL)

Consider this proposition in first-order logic: For any interpretation $I$, any closed formula $\phi$ and any two valuations $\rho$, $\sigma$. $I\rho \models \phi \iff I\sigma \models \phi$ This is ...
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Countable transitive model of ZFC and $\mathcal{P}(\omega)$

Let $\mathbb{M}$ be a countable transitive model of ZFC. I understand that $\omega^\mathbb{M} = \omega$ but $\mathcal{P}(\omega)^\mathbb{M} \neq \mathcal{P}(\omega)$, for ...
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Model Theory - Equivalence of formulas using automorphisms

Let $\mathbf Q$ denote the additive group of rational numbers, i.e. the structure $\mathbf Q = (\mathbb Q;+,0)$. Let $L$ be the language of $\mathbf Q$ and let $T$ be the complete theory of $\mathbf ...
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Why are $n$-ary function symbols interpreted as $n$-ary operations and not general $n$-ary functions?

I would appreciate if someone here could check the correctness of my reasoning about the following. I'm new to logic. Let $\mathfrak{M} = \langle M,\mathfrak{I} \rangle$ be a structure for some ...
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Ultraproducts and Elementary Embeddings

Let $K= \{A_i: i\in \omega\}$ be a countable collection of $L-$structures. Suppose that for each $A_i, A_j$ in $K$, $\exists A_p \in K$ such that $i,j< p$ and $A_i \prec A_p $ and $A_j \prec ...
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Is this an accurate description of structures and interpretations.

I read about structures and interpretations today. I've described them below this paragraph. Have I accurately described them? If not, what have I incorrectly described? A structure, $\mathscr{A}$, ...
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49 views

Let $T$ be the theory of linear, dense order, without minimum or maximum. Is $T\cup\{c_{i}<c_{j}\mid i<j\}$ complete?

Let $T$ be the theory of linear, dense order, without minimum or maximum in the language $\mathscr{L}$ . Expend the language by adding it countable amount of constants: ...
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33 views

Quick question about the relation between elementary classes and pseudo-elementary classes

Let $\mathcal{L}$ be a logic and $\mathscr{K}$ a class of structures in the vocabulary of $\mathcal{L}$. We say that $\mathscr{K}$ is a (basic) elementary class iff there is $\phi \in \mathcal{L}$ ...
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69 views

$T$ be a theory that claims that both $P$ and $\neg P$ are infinite. Prove that $T$ is a complete theory

Let $\mathscr{L}=\{P\}$ a language with one unary predicate. Let $T$ be a theory that claims that both $P$ and $\neg P$ are infinite. Prove that $T$ is a complete theory. I tried to prove by ...
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148 views

Unexpressibility of a property in first order logic

We can give a very general notion of what is to iterate a function. Given a set $\mathcal U$ and a function $f:\mathcal U \rightarrow \mathcal U$, then, to iterate the function $f$ will mean to ...
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36 views

Models of $T_\exists$ are precisely the structures that contain a model of $T$

An usual exercise in introductions to model theory is to prove the following statement: Let $T$ be a theory, and let $T_\forall$ be the set of all universal sentences that are consequences of $T$. ...
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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) ...
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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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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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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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40 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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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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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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90 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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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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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 ?
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41 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 ...
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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 ...
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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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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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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
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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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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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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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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 ...
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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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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 ...
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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 ...
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1answer
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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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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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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), ...
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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 ...
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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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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 ...
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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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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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$n$-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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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 ${\cal B}$ ...
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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 ...