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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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 ...
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34 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 ...
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38 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 ...
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51 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 ...
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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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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 ...
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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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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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164 views

Does a finitely axiomatizable theory with $\Sigma_n$ axioms have $\Sigma_n$ theorems?

Let $T$ be a theory with a finite set of axioms $\Delta$, where every sentence of $\Delta$ is $\Sigma_n$ (in the Levy hierarchy). Is every theorem of $T$ (i.e. every sentence that is proved by $T$) ...
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What are all kind of “metamath” good for? Can it help me here?

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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58 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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43 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 ...
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3answers
65 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 ...
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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 ...
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52 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 $A$, ...
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79 views

Completness in higher order logic and Interpretations

It´s known that for first order theories, it holds $\mathbf{ZFC} \vdash T \vdash \varphi \leftrightarrow T \models \varphi$. Why does this not hold in the higher order case (any simple example?)? ...
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51 views

In modal logic, why are models ordered sets?

I just started undergrad math, so I only have a fuzzy idea of what a model is. I'm learning about modal logic in one of my classes. Our text describes modal logic as operating in a model defined as an ...
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103 views

Is $\sqrt{2}$ definable in $(\mathbb{R},+,\cdot,0,1)$?

For an exercise in model theory I have to state if $\sqrt{2}$ is a definable element of the structure $\mathcal{R}=(\mathbb{R},+,\cdot,0,1)$. I expect it is not, but I haven't been able to prove this. ...
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49 views

Question from Hodges' textbook Shorter Model Theory

I'm thinkng through Section 2.3. of Hodges' textbook Shorter Model Theory, problem 7(b): "Let $L$ be a first-order language. Show (without assuming that every structure is non-empty ) that every ...
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Axiomatizability of the multiplication of a ring

The operation of ring multiplication is axiomatizable, if we allow ourselves an additional auxillary addition symbol. Just write down the ring axioms in the signature $\{*,+\}$. But could ...
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1answer
48 views

Is division axiomatizable?

Consider a set $G$ with a group operation. We can define a division operation $a*(b^{-1})$ and call it $\operatorname{div}$. Is the class of division operations first order axiomatizable? And if so, ...
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For a complete truth-set $T$ is a countable transitive model satisfying $T$ unique?

Let $T$ be a maximal (in the sense that either $\phi \in T$ or $\phi \not \in T$ for all $\phi \in \mathcal{L}_\in$) set of sentences consistent with $ZFC$. Question For a countable transitive model ...
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54 views

You don't need to take an algebraic closure twice in model theory

This is an exercise (1.4.11) from Marker. Fix a language $\mathcal L$ and $\mathcal L$-structure $\mathcal M$. For a subset $A \subseteq M$, an element of $M$ is algebraic over $A$ if it is a member ...
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40 views

Can a binary relation on a set $S$ isomorphically embed every binary relation on $S$?

Is there any binary relation $R$ on a non-empty set $S$ such that $R$ isomorphically embeds every binary relation on $S$? (By "$R$ isomorphically embeds $Q$" I mean: there is a one-to-one function ...
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81 views

Prove that $\Pi_{i \in I} \mathbf{A}_i \cong \Pi_{j \in J} (\Pi_{i \in I_j} \mathbf{A}_i)$ where $\{I_j \ | \ j \in J\}$ is a partition of I.

My problem is following: prove that $\Pi_{i \in I} \mathbf{A}_i \cong \Pi_{j \in J} (\Pi_{i \in I_j} \mathbf{A}_i)$, where $\langle \mathbf{A}_i \ | \ i \in I \rangle$ is an indexed set of similar ...
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Show that every element in class $\mathcal{K}$ have at most $n$ elements

Suppose that $\mathcal{K}$ is a class of finite structure of language $\mathcal{L}$. If $\mathcal{K}$ is axiomatizable then prove that exist $n$ such that every structure from $\mathcal{K}$ have at ...
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51 views

Why isn't there a first-order theory of well order?

Problem 1.4.1 of Model Theory by Chang and Keisler asks, Is there a theory of well order in the first-order language $\{\leq\}$? I'm pretty sure the answer is no, since well order is a property ...
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23 views

PA can define 6's multiplier?

Let set A be : {6, 36, 216, 1296 .....} i.e. A={ $6^k$} where $k \in \mathbb{n} $ In the Model PA, can PA define set A? I know PA can define set { $2^k$} and set { $3^k$}. However what about { ...
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Types realized in ultrapowers consisting of definable functions

Let $\mathcal{M}$ be a nonstandard model of arithmetic and let $M$ be its universe. Let $U$ be a nonprincipal ultrafilter over $M$ and let $\mathcal{N}$ be the ultrapower $\mathcal{M}^M / U$. Let $F$ ...
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172 views

Difference between completeness and categoricity

I have problems understanding the difference between a categorical theory and a complete theory. My intuition says that every valid complete theory must be categorical. Is it true? Clarification: by ...
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1answer
91 views

Indiscernible subsequences

Work in a saturated model $\cal U$ of sufficiently large cardinality. Are there assumptions on $\kappa<|\cal U|$ that guarantee that any sequence $\langle a_i:i<\kappa\rangle$ has an ...
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1answer
28 views

Adding unary relation symbol within complete theory

I try to prove following problem: Let $T$ be a complete theory over countable language, then $T$ has a model $\mathfrak{A}$ with cardinality $\le 2^{\aleph_0}$ such that for each $\mathfrak{B}\models ...
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2answers
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Is $Th(\mathbb{Z}[x])$ uncountably categorical?

Consider $T=Th(\mathbb{Z}[x])$ in the language $L = \{0,1,+,\times,deg(), \circ\}$ where $0,1,+$ and $\times$ have their usual interpretations, $deg()$ is a unary function symbol which gives the ...
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Are ideals necessarily definable?

Consider the first-order language of rings. Let $R$ be a ring and $I \subseteq R$ be an ideal. Is $I$ necessarily $\emptyset$-definable? If not, what if we allow parameters from $R$?
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How to prove an element of a given structure is not definable?

Let $A$ be the set of all $q$ in $\mathbb{Q}$ such that $q\leq0$ or $1\leq q$, and let $\mathcal{A}=(A,<)$ be a structure. I have to show that 2 is not a definable element of this structure, e.g. ...
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Proving that the two structures $(\mathbb{R},<)$ and $(\mathbb{R}\setminus\{0\},<)$ are not isomorphic

For an exercise on model theory I have to prove that the structure $(\mathbb{R},<)$ is not isomorphic to $(\mathbb{R}\setminus\{0\},<)$. I found the function $f(x)=\begin{cases} x & ...
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3answers
118 views

Defining finite unions of intervals with algebraic endpoints on the reals

I'm currently working a bit on Enderton's logic textbook (2nd ed), and, on the second chapter, he marks the following exercise on definability with an asterisk. Let $(\mathbb{R}; +, \cdot)$ be the ...
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Local embedding implies embedding into an ultraproduct

I am reading Gorbunov's "Algebraic theory of quasivarieties" and can't prove some statements, which are supposed to be obvious I think. At first, here are some definitions and notations. For a given ...
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Show that exists a finite subset in $\mathcal{L}$-theory

Let $\mathcal{L}$ be a language and let $T$ and $T^{\prime}$ be $\mathcal{L}$-theories. Suppose that for every model $\mathcal{M}$ of $T$ there exists $\sigma \in T^{\prime}$ such that $\mathcal{M} ...
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Proving that a propositional theory of any cardinality has an independent set of axioms

This is exercise 1.2.19 from Chang & Keisler's Model Theory, which has been giving me a headache for some time now. Let $\mathscr{S}$ be a given propositional language of any cardinality (i.e. ...
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complete, finitely axiomatizable, theory with 3 countable models

Does it exist a complete, finitely axiomatizable, first-order theory $T$ with exactly 3 countable non-isomorphic models? A few relevant comments: There is a classical example of a complete theory ...
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Is it meaningless to say $M\prec N$ for two proper class models?

Kunen in page 88 of his "Set Theory" book says: ... For a specific given $\varphi$, the notion $M\prec_{\varphi}N$ (i.e. $\forall \overline{a}\in M~~~M\models \varphi ...
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Are there counter intuitive interpretations of ZF or ZFC?

Are there interpretations of ZF or ZFC that are non standard in the sense that $\epsilon$ is interpreted in a counter intuitive way that intuitively has nothing to do with "belongs to" or "is part of" ...
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Can one prove existence of incommensurables without the Pythagorean theorem?

Euclid's proof that the side and the diagonal of a square have no common measure, probably going back to Pythagoreans, reduces it to proving the irrationality of $\sqrt{2}$. This reduction uses the ...
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Exercise 4.5.13 in Marker

I am solving exercises in Model Theory: An Introduction from David Marker. So far I didn't get anywhere with the second part of the following exercise: Exercise 4.5.13 Let $\Delta$ be a set of ...