# Tagged Questions

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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### 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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### 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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### 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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### 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 ...
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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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### 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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### 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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### 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 3.1.4:...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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} \...