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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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 ...
2
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2answers
32 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 ...
2
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0answers
52 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 ...
1
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2answers
27 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 ...
5
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1answer
40 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 ...
0
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1answer
18 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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0answers
55 views

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 $F$ ...
2
votes
3answers
158 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 ...
7
votes
1answer
82 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 ...
2
votes
1answer
24 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 ...
3
votes
2answers
60 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 ...
2
votes
2answers
74 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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1answer
27 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
42 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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0answers
106 views

Is the axiom of choice constructive in the constructible universe?

Even ZF has some non-constructive elements mostly due to contradiction proofs. For example one may be able to construct a sequence of objects some of which have a given property without being able to ...
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3answers
84 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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0answers
43 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
votes
1answer
39 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} ...
1
vote
1answer
62 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. ...
5
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0answers
87 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 ...
4
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2answers
79 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
361 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" ...
3
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2answers
119 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 ...
2
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1answer
43 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 ...
3
votes
2answers
80 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 ...
4
votes
2answers
58 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
votes
1answer
74 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
votes
1answer
35 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 ...
2
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0answers
49 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 ...
3
votes
1answer
103 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
votes
1answer
104 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, ...
2
votes
1answer
118 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 ...
0
votes
0answers
73 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 ...
3
votes
2answers
69 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 ...
2
votes
1answer
24 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 ...
2
votes
2answers
96 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 ...
1
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1answer
59 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 ...
0
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28 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 ...
3
votes
0answers
26 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 ...
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1answer
48 views

Consistent Set of Sentences is Consistent in Expanded Language

Suppose that we have a set $\Phi$ of sentences over a first-order language $\mathcal{L}$ and that $\Phi$ is consistent. Suppose we have another first-order language $\mathcal{L}'$ such that ...
5
votes
0answers
85 views

ind-completion and functors which are full with respect to isomorphisms

Let $C$ and $D$ be categories and $F:C\rightarrow D$ a faithful functor which is full with respect to isomorphisms. This means that if $a,b\in C$ and $f:F(a)\rightarrow F(b)$ is an isomorphism in $D$, ...
0
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3answers
94 views

a theory $(x,+,\cdot)$ satisfies $x \cdot x=0$ and $x \cdot (y \cdot z)=(x\cdot y)\cdot z$ [closed]

I asked the question a theory that satisfies $x \cdot x=0$ and $x \cdot (y \cdot z)=(x\cdot y)\cdot z$, but I lost my account since then. So. I am posting my edit as a new question. Let there be a ...
2
votes
1answer
36 views

Compactness and Arithmetic Confusion

Let $T$ be some theory capable of arithmetic and construct a provability predicate (which we will call $Prb_T$). Let $\mathbb{N} \models T$. Expand our language to include a new constant symbol $c$. ...
6
votes
1answer
107 views

Using the compactness theorem to disprove axiomatizability

Another model-theoretic exercise from Smirnov's book. Problem: Construct infinite family of varieties such that their union is not axiomatizable. My solution: Denote by $\mathcal{A}_n$ the variety ...
3
votes
1answer
68 views

Best algebra text for Model Theory

I'm looking for an algebra book that is tailored towards some of the ideas in Model Theory, I'm currently slogging through Hodges' Model Theory. I'm a bit rusty with my algebra and was curious if ...
4
votes
1answer
76 views

How can I imagine a model of $\mathsf{(ZF-Inf)} + \neg \mathsf{Con(ZF-Inf)}$?

Gödel's second incompleteness theorem states that if $\mathsf{ZF-Inf}$ is consistent, then $\mathsf{ZF-Inf} \nvdash \mathsf{Con(ZF-Inf)}$. Moreover, if $\mathsf{(ZF-Inf)} + \neg \mathsf{Con(ZF-Inf)}$ ...
3
votes
1answer
60 views

real closure of an archimedean field

my question is: Is an archimedean field dense in its real closure? I know that in the non-archimedean case, this does not have to be true (e.g., rational fucntions). Thanks!
2
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2answers
90 views

Uncountable models for integers

Part of Asaf Karagila's brilliant answer to one of my other questions puzzles me a lot. Namely, I find it hard to understand how there can be a model for ZFC with uncountably many integers. My ...
2
votes
1answer
91 views

Maximal model for $\Bbb R$?

I have not dealt professionally with set theory, so excuse me if my way of formulating this question does not completely follow standard terminology. Actually, my question is about whether or not the ...
3
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1answer
77 views

Relations between Theories and Categories

I'm just toying around with some thoughts, trying to grock some concepts: It seems that every formal theory induces a locally small category via interpretations: its objects are structures that ...