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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Regarding the theory $REI_{\alpha}$

The theory $REI_{\alpha}$ has as its language $L=\{E_{\beta}|\beta\leq\alpha\}\cup{\{E_{-1}\}}$, and each $E_{\beta}$ and $E_{-1}$ are binary relation symbols. Let $T$ (=$REI_{\alpha}$) be the theory ...
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29 views

Prove that formule is true in every realisation of language [duplicate]

I have language $\mathcal L$= $\{$P,Q,R$\}$, arity of $P,Q,R$ is $1,1,2$. And I have two formulas: 1) ($\forall x$ $P(x)$ $ \land $ $\forall x$ $Q(x)$) $\leftrightarrow$ $\forall x$ ($P(x)$ $ \land $ ...
3
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0answers
61 views

Positive existential theory of an extension of the ring

When we know that the positive existential theory of a ring $R[x]$ in a language $L$ is undecidable, does it follow that the positive existential theory of $R[x,y]$ in the same language $L$ is also ...
3
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1answer
34 views

Parameters and strongly minimal sets

Suppose $T$ is a countable complete theory, with monster model $\mathbb{C}$. A definable set $D := \phi(\mathbb{C}, \overline a)$ is strongly minimal if given any other formula $\psi(x, \overline b)$ ...
2
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38 views

How to show $T=\{\forall x(x\neq S^{n}x)|1\leq n\}\cup\{\sigma\}$ in $\mathcal{L}=\{S\}$ is $\kappa$-categorical

Problem saying: Let $T=\{\forall x(x\neq S^{n}x)|1\leq n\}\cup\{\sigma\}$ in $\mathcal{L}=\{S\}$ , where $S$ is a unary function and $S^{n}$ abbreviates $\underbrace{S\dots S}_{n}$ , and ...
5
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1answer
159 views

Can there be a countable transitive model satisfying the same $MK$ theory as $V$?

A little while ago, I asked whether or not there could be a countable transitive model satisfying the same $ZFC$ theory as $V$ (assuming that we're working within some $V$, or (if you like) that there ...
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1answer
45 views

Defining a formula using FO+TC

Define a signature Σ and an FO+TC formula ϕ over Σ, such that: there is no infinite structure satisfying ϕ for every even natural number n>0 there is a structure of size n satisfying ϕ for every ...
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46 views

model theory & algebraically closed field

recently I'm taking mathematical logic course, and the class covers some basic model theoretical ideas. Since I have not taken any abstract algebra course, It is so hard to understand what is going ...
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1answer
40 views

How should one read the s*(t) function in Mendelson's Introduction to Mathematical Logic?

I'm self-teaching logic and doing it by means of following Elliot Mendelson's Introduction to Mathematical Logic (6th edition). In p.56 he defines a a function s* which, in his words, 'assigns to each ...
6
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168 views

Countable elementary submodels

I'm having some trouble understanding elementary submodels. Let $H_\chi$ be the set of all sets which are hereditarily of cardinality $<\chi$. Let $\textbf{N}=(N,\in)$ be a countable elementary ...
4
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1answer
35 views

Why in countably saturated models, types that are consistent with $TH(\mathcal{M_a})$ are finitely realizable?

I'm learning about countably saturated ($\alpha$-saturated) models. There is a hidden presupposition everywhere used: The type $\Gamma(x)$ is consistent with $TH(\mathcal{M_a})$ iff $\Gamma(x)$ is ...
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1answer
47 views

Is model theory needed to understand ordinal logic?

By ordinal logic, I mean turing's ordinal logic. I'm going to learn first order logic, elementary set theory, basic computability, and godelian incompleteness as prerequisites for ordinal logic. But, ...
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20 views

Let $L$ be a first order language with equality and two binary function symbols…

$\bf Question$ Let $L=(F=\{f^2,g^2\}, P=\{=\}, C=\{c\})$ be a first order language. Let $I = (\Bbb R^{2\times 2}, f^2(x,y)=(x+y)^t,g^2(x,y)=x+y,0) $ be an interpretation of $L$. Exhibit ...
4
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1answer
163 views

Can't EF game theory be applied to finite languages WITH function symbols?

Let $\mathcal{M}$ and $\mathcal{N}$ be two structures in a language $\mathcal{L}$. We define the finite determined game $G_n(\mathcal{M},\mathcal{N})$ as a game with $n$ rounds where in each round ...
2
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1answer
34 views

non-isomorphic countable models of $Th(\mathbb{N})$

I'm proving there are exactly $2^\omega$ non-isomorphic countable models of standard natural numbers. I got cardinality of them $\geq 2^\omega$from prime arguments. but I don't get how to prove other ...
2
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1answer
37 views

Is this an axiomatization of real closed fields?

I know that real closed fields are defined as ordered fields where every positive element is a square and every odd polynomial has a root. But can they also be axiomatized as totally ordered fields ...
3
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1answer
65 views

The analogy between two Rudin-Keisler orders

Given a set $X$, an ultrafilter $U$ on $X$, and a function $f\colon X\to Y$, we can push forward $U$ along $f$ to obtain an ultrafilter $f_*U$ on $Y$, defined by $C\in f_*U$ if and only if ...
5
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115 views

Equivalence of the theories $\operatorname{Th}(\Bbb{R}, 0,1,+, \le)$ and $\operatorname{Th}(\Bbb{Q}, 0,1,+, \le) $

So I was working on showing that $$\operatorname{Th}(\Bbb{R}, 0,1,+, \le) = \operatorname{Th}(\Bbb{Q}, 0,1,+, \le) $$ My initial idea for working on this problem was to systematically start by ...
3
votes
1answer
49 views

Axioms for $\mathbb Z$-groups without named one?

The theory of $(\mathbb Z,+,0,1)$ has been studied as the theory of $\mathbb Z$-groups, and it has been examined as a series of exercises (and I'm sure other places) in David Marker's Model Theory ...
2
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1answer
52 views

The Amalgamation Property in AECs

I am new to abstract elementary classes (AEC) and am reading the early chapters of Baldwin's Categoricity text. I have shown that the disjoint amalgamation property holds for an AEC obtained by ...
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1answer
49 views

Is a tautology substitution instance with first order formulas valid?

I wonder how to show the following: Let $P_1,...,P_n$ be propositional symbols occurring in a tautology $\alpha$. Assume that $\varphi_1,...,\varphi_n$ are first order formulas and that $\alpha'$ ...
4
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1answer
119 views

Using Compactness to find a non-constructible set

$\newcommand{\ZFC}{\mathit{ZFC}}$I was trying to explain the first ideas of forcing to a friend and I recalled the construction of a model of non-standard arithmetic by using compactness. It is clear ...
6
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1answer
97 views

How do inconsistent arithmetics get around Gödel incompleteness?

Gödel theorems imply that arithmetic can not be complete under recursive axiomatization and consistency assumptions. Unexpectedly, consistency can be dropped in a meaningful way by switching to a ...
2
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1answer
91 views

Understanding countable elementary submodels

So I'm having some trouble understanding the existence of countable elementary submodels. I have read and understand the Löwenheim–Skolem theorem, so given a model I understand how to build a ...
2
votes
1answer
74 views

Is the use of the meta-meta-theory allowed in proving an independence result?

I am wondering about the use of a meta-meta-theory in proving statements of the form "$\phi$ is independent of $\mathcal{T}$" in mathematical logic. Short question: Is using a result from the ...
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1answer
40 views

Simple proof that Automorphisms preserve definable subsets?

I've been looking all over for a proof of this result, but I haven't really found anything. Is anyone aware of a particularly simple or elegant one?
2
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1answer
58 views

Why isn't $\leq$ definable in $(\mathbb{R};0,+,-)$?

Are there any simple and straightforward proofs of this fact? I'm not really sure how to begin to approach the problem.
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40 views

Automorphisms and Definable subsets?

I'm trying to show that if we have a structure $\textbf{A}$ and $C\subseteq A$, with an $\mathcal{L}_C$ formula $\varphi(x)$ defining the set $S$ in $\textbf{A}$, then for any given automorphism ...
0
votes
1answer
34 views

Is logical implication examining only the syntatical component?

As my title asks, is logical implication only examining the syntatical compinent of the formal language? I am using Enderton's book on Mathematical Logic in my class and after some work I am ...
3
votes
1answer
52 views

Show the inequality $x <y$ is definable in the language $\langle \mathbb{R}; +, \times ; 0,1 \rangle $

My initial idea is that I need to find a sentence that expresses 'x is positive' and then I can say: for any $a, b$, $a>b$ iff there is a positive x s.t. $b+x=a$, but can't figure out how, any ...
3
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1answer
48 views

A simple proof that elementary equivalence and isomorphism coincide for finite structures?

I'm wondering if there's a straightforward proof of this result I've seen mentioned in quite a few places. If $\mathcal{L}$ is finite, of course, this is trivial since there's a single formula that ...
1
vote
1answer
54 views

First-order definition of “$f$ is continuous at $x$” using just $<$

I need to show that the set $\{ a\in \mathbb{R}\ |\ f\ \text {is continuous at a}\}$ is definable in the structure $(\mathbb{R};<,\ f)$, where $f$ is just some function ...
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1answer
35 views

Prove that this theory is incomplete

Given $\Sigma = \{\forall x \forall y \forall z(x \circ (y \circ z) = (x \circ y) \circ z), \forall x (x \circ e = x), \forall x \exists y (x \circ y = e)\}$ (eg. group theory axioms), I need to prove ...
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71 views

Prove that there exists a non-standard model of arithmetic

Specifically, I need to show there exists a structure $\mathcal{Z}'$ that is elementarily equivalent to $ \langle \mathbb{N} ; <; \cdot ; 0,1 \rangle $ (the standard model of arithmetic) but is not ...
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1answer
39 views

Relation between homogeneity and categoricity

Define a structure $\mathcal{A}$ as homogeneous if, for every two substructures $\mathcal{B}, \mathcal{C}$ of $\mathcal{A}$, if $f: B \to C$ is an isomorphism, then $f$ extends to an automorphism of ...
3
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1answer
62 views

Is a theory elementary iff it is categorical? [closed]

If an elementary class is the set of structures satisfying a theory, and a theory is categorical if it determines a structure up to an isomorphism, it would seem that the two concepts are related, no? ...
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56 views

bi-interpretability and automorphism groups

Let $M$ and $N$ be two first order structures, say they are countable and $\aleph_0$-categorical. Then $M$ and $N$ are bi-interpretable if and only if their automorphism groups $Aut(M)$ and $Aut(N)$ ...
2
votes
1answer
60 views

Show the only definable subsets of $ \langle \mathbb{Q} , < \rangle $are the empty set and $ \mathbb{Q} $

I'm trying to show the above, and I know it should be quite simple, but I'm struggling to get my head around how you show subsets are not definable using automorphisms- so a detailed explanation would ...
2
votes
1answer
46 views

DOP, Shelah, Clarifying the definition

I'm trying to understand Shelah's prototypical example of DOP here on page 26, item (c). It would be a very nice example if correctly understood.However, there is some strange mixture in the ...
6
votes
1answer
105 views

Integer parts isomorphic?

Let $F$ be a real closed field. It is known$^{[1]}$ that $F$ has an integer part, that is, a subring $A$ such that $\forall x \in F, \exists ! a \in A, a \leq x < a+1$. Are all integer parts over ...
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89 views

Calculus as a structure in the sense of Model theory

I am not a specialist in Logic (my field is Functional Analysis), so excuse me my ignorance. I suppose there must be texts where Calculus is presented as a structure in the sense of Model theory. I ...
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0answers
45 views

Quantifier elimination in the structure of exponential sums

We consider the language $L=\{+, -, ' , T, 0, 1\}$ Let $\text{Exp}(\mathbb{C})$ (the exponential sums) be the structure in that we interpret $L$. We define $\text{Exp}(\mathbb{C})$ as the set of ...
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2answers
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Exercise 1.2.10 from Model Theory by Chang and Keisler.

I am reading Model Theory by Chang and Keisler, and I am having some trouble with exercise 1.2.10, which asks me to prove that if $\Sigma \vdash \varphi$ for all $\varphi \in \Gamma$ and $\Sigma \cup ...
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1answer
40 views

restriction map in Stone space is open

In the Stone space, how to proof that the restriction map $S_{m+n}(B)\to S_{n}(B)$ is open? Where B is subset of the model of the theory. I know that the restriction is continuous and surjective.
4
votes
1answer
135 views

Is there a countable transitive model satisfying the same set of first-order sentences as $V$? [duplicate]

This is probably a pretty simple question, but I'm tying myself in knots over it. We're all familiar with the Reflection Theorem, Lowenheim-Skolem Theorem, and Mostowski Collapse Lemma for getting ...
2
votes
1answer
46 views

Proving that every interval in an o-minimal structure is definably connected.

By an ordered structure I mean a (first order) structure $\mathcal{M}=(M,<,\ldots)$ that is totally and densely ordered by $<$. An ordered structure $\mathcal{M}$ is o-minimal if every definable ...
3
votes
1answer
68 views

Using the compactness theorem to prove Principal ideal rings nonaxiomatizable

I'm trudging through Barwise's Handbook of Mathematical Logic and came across this: Here is a good exercise. A ring $\mathfrak{R}$ is a principal ideal ring if $\mathfrak{R}$ is a model of the ...
2
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0answers
25 views

$R \subseteq M^I$ is $A$-invariant, then $R$ is $A$ definable in the next two infinitary logics.

Let $A \subseteq M$ and let $R \subseteq M^I$ be $A$-invariant. If $M$ is $\kappa^+$-strongly homogenous for $\kappa = |A| + |I|$, then $R$ is $A$-definable in $M$ in the infinitary logic ...
2
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1answer
51 views

For $\mathbb{N}$ a structure in $\mathcal{L}=\{s, 0, 1\}$, are the sum and product definable? [duplicate]

Reading about interpretability made me think about the $s$ function that somehow always bothered me in the language of PA. My question is the following Given $\mathbb{N}$ as a structure in ...
0
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1answer
43 views

Showing that a certain formula of second order logic with full semantic is true in all and only non-standard model of arithmetic.

Showing that a certain formula of second order logic with full semantic is true in all and only non-standard model of arithmetic. $\exists X(\exists x Xx \wedge \forall x\forall y((Xx \wedge (x=0 ...