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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140 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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115 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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74 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
143 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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1answer
262 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 ...
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1answer
60 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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107 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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1answer
68 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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1answer
137 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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1answer
97 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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72 views

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
58 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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112 views

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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98 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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2answers
60 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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94 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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44 views

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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1answer
181 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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1answer
27 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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3answers
391 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
119 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
59 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
81 views

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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127 views

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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63 views

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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2answers
112 views

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
263 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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1answer
129 views

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 ...
2
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1answer
46 views

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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1answer
152 views

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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446 views

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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2answers
113 views

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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4answers
421 views

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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2answers
165 views

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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1answer
93 views

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 ...
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100 views

Suppose $R \sim_\omega R'$. Then for every $k$-tuple $a$ in $E$ and every natural number $p$, there is a $k$-tuple $b$ in $E'$ such that $a \sim_p b$

Sorry to bother you guys again with a Poizat question, but I'm struggling a little bit with the material (as it must be obvious) and I want to check if I got the main idea correctly or if I'm totally ...
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2answers
232 views

Proper definition of quantifier elimination

I study Marker book "Model Theory, An Introduction". Definition 3.1.1 on page 72 defines "theory T has quantifier elimination". A theory $T$ has quantifier elimination if for every formula $\phi$ ...
4
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1answer
139 views

Elementary Model Theory

I'm working through section 4.3. on model theory from Dirk van Dalen's Logic and Structure (fifth ed.) and am struggling with van Dalen's sometimes sloppy way of presenting proofs. As usual let a ...
2
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1answer
46 views

Equality of sets of local isomorphisms between relations

I'm still working on the first pages of Poizat's A Course in Model Theory. I'll state the basic definitions again, in order to avoid referring back to an early question: Poizat defines an isomorphism ...
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81 views

On the back and forth conditions for a set of partial isomorphisms

I've recently begun reading Poizat's A Course in Model Theory and already in the first pages I had some doubts. One odd (not necessarily bad) thing is that he defines notions such as isomorphism only ...
4
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1answer
180 views

Why is the powerset axiom more acceptable than the axiom of choice?

The key step in Zermelo's proof of the well ordering theorem is to use $\text{AC}$ to simultaneously choose the next elelment for all possible partial chains in prospective well orderings, but that ...
7
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1answer
166 views

Nonstandard complex numbers and categoricity

Let ${}^*\mathbb{C}$ be a nonstandard complex number field (given, for instance, as a countable ultrapower.) By the transfer principle ${}^*\mathbb{C}$ is algebraically closed of characteristic zero, ...
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1answer
175 views

Keisler Order: Saturated Ultrapowers

Keisler's paper "Ultraproducts which are not Saturated" states the following theorem as a corollary to a (much more) generalized theorem. However, I cannot figure out how to prove it for the specific ...
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115 views

What is even meant by the “cardinality of a model?”

Please help me understand even the most basic ideas in model theory: When in model theory we speak of the cardinality of a model, what exactly is meant by that? I assume that when we say that the ...
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2answers
120 views

Clarifications on proof of compactness theorem

I've been reading through the following proof of compactness theorem: http://www.princeton.edu/~hhalvors/teaching/phi312_s2013/compactness.pdf One thing that struck me is that this proof seems to ...
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1answer
34 views

Density and Saturated Models.

Consider $(\mathbb{Q}; \leq)$ and let $T$ be the theory of dense linear orderings without endpoints. Let $\mathfrak{A}$ be the $\omega_1$ saturated model of $T$. Note that ...
3
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2answers
126 views

Countable structure with qe and not ultrahomogeneous

Here: The connection between quantifier elimination, $\omega$-categorical and ultrahomogenous I gave an example of an uncountable structure, that is not ultrahomogeneous but has quantifier ...
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1answer
134 views

Are order isomorphic real closed fields isomorphic?

There are counterexamples to order isomorphisms of ordered fields being field isomorphisms, see Is the multiplicative structure of a totally ordered field unique?. However, Wikipedia suggests that for ...
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53 views

When is the higher-order theory of a model categorical?

I'm interested in (classical) type theories $L$ with the following property. For $M$ any model of $L$ (in Set), let $T(M)$ be the type theory of $M$, i.e., the strongest extension of $L$ satisfied by ...
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1answer
49 views

Let $h: A \to B$ be a weak homomorphism. Is h$[A]$ a substructure of $B$?

A little bit more precise: let $\mathfrak{A}$ and $\mathfrak{B}$ be two structures. Define a weak homomorphism as a function $h: \mathfrak{A} \to \mathfrak{B}$ such that the folowing conditions are ...