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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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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2answers
94 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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42 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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53 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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169 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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64 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 ...
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1answer
44 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
76 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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108 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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84 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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376 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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131 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
54 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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2answers
87 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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83 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
88 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
37 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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51 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
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1answer
114 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 ...
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1answer
111 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
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1answer
128 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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75 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
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2answers
78 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
27 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 ...
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99 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
67 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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36 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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32 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
53 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 ...
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107 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 $D$, ...
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3answers
98 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
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1answer
37 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
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1answer
121 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
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2answers
99 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
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1answer
89 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
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1answer
62 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!
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2answers
100 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
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1answer
94 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
86 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 ...
4
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1answer
39 views

When a subgroup of automorphism group of a structure is in the form of automorphism group of a substructure?

Question 1: Is the following statement true? ($*$) Let $\mathcal{L}$ be a first order language and $\mathcal{M}$ a $\mathcal{L}$-structure and $H\leq Aut(\mathcal{M})$ then there exists a ...
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1answer
166 views

Is “PA has no non-standard models” consistent with ZF?

I have seen several proofs that there exist nonstandard models of arithmetic, but they all seem to rely on the compactness theorem, which is not implied by ZF. So are there any proofs in ZF that ...
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1answer
54 views

A question about second-order logic and inaccessible cardinals.

Let $\kappa$ denote an inaccessible cardinal, and suppose $T \in V_\kappa$ is a second-order theory. Now consider some mathematical structure $X \in V_\kappa$. Then I think it is clear that $X \models ...
3
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1answer
45 views

Elementary equivalence of models

I'm quite new to model theory, so please correct me if I'm using wrong terminology. I need help with an exercise from Smirnov's book "Varieties of algebras" (In Russian). Problem: Assume that a ...
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1answer
35 views

Relative Interpretations alla Kunen

at the moment I try to figure out some details of Kunen's "Relative Interpretation" Definition (within the 2013 Edition of his "Set Theory", p. 99 to 100): Definition If $\Lambda$ is some axioms ...
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1answer
70 views

Decision and the Uncountable Spectrum

In 2000, Hart, Hrushovski, and Laskowski classified all complete first order theories in a countable language up to their uncountable spectra. However, does this also imply that given a $any$ ...
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1answer
67 views

Are algebraic structures required to satisfy axioms?

Is it a requirement for algebraic structures, when studying universal algebra, to satisfy axioms? The reason I ask is because algebraic structures are only defined by a underlying set, a signature, ...
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1answer
129 views

Elementary equivalence of free groups

This must be known inside out by model theorists by I have no cluse whether the following is true or not: Denote by $F_n$ the free group on $n$ generators. Suppose that $n\neq m$. Are the groups ...
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91 views

An Uncountable language , A Model of $\mathbb{N}$, A Problem.

Edit 1: I messed up my original question, but Arthur Fischer answered my question anyway. Edit 2: We can actually restrict $L$ to the language in arithmetic with the prdicate $P_{\mathbb{P}}$. ...
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3answers
262 views

Upper bound on cardinality of a field

Is there an upper bound on the cardinality of a field? The "biggest" fields I know are the field of real numbers, or the field of complex numbers. Is there a field with cardinality greater than ...
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1answer
80 views

Is infinitary logics $\mathcal{L}_{\infty\omega}$ an abstract logic?

Infinitary logics $\mathcal{L}_{\infty\omega}$ is an extension of first-order logics such that $\bigvee\Phi \in \mathcal{L}_{\infty\omega}$ if $\Phi$ is a set of ...