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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How to show the relation $<$ is not definable in $(\Bbb N; 0, \operatorname {S})$ by quantifier elimination?

Show that the ordering relation $\{(m, n)| m < n \in \Bbb N\}$ is not definable in $\mathfrak{N}_{s}$. Suggestion: It suffices to show there is no quantifier-free definition of ordering. ...
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271 views

Axiom Systems and Formal Systems

I'm a really beginner in Mathematical Logic.I'm currently reading Shoenfield Mathematical's Logic and i'm having a hard time trying to relate the concept of Formal Systems with the concept of Axiom ...
2
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1answer
81 views

Compactness principle via model theory.

A standard method of getting more concrete results from more abstract ones in Ramsey theory is the so called Compactness Principle. It is best illustrated by example. Here is the standard version of ...
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1answer
188 views

Questions about the concept of Structure, Model and Formal Language

When we start to define mathematical logic (specifically, propositional, first order, and second order logic) we start defining the concept of a language. At the begining this is done in a purely ...
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115 views

the amalgamation property

Definitions: The age of a structure $M$ is the class of finitely generated substructures of $M$. A class of structures K has the amalgamation property (AP) if Whenever $A,B,C$ belong to $K$ ...
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340 views

Is the compactness theorem (from mathematical logic) equivalent to the Axiom of Choice?

Or more importantly, is it independent of the axiom of choice. The compactness theorem states the given a set of sentences $T$ in a first order Language $L, T$ has a model iff every finite subset of ...
3
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1answer
142 views

Model theory/stability theory

I was flipping through Baldwin's Stability Theory book and saw an example that has me confused... The example is a 1st-order theory $T$ of refining equivalence relations $E_i(x, y), i< ω$, where ...
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2answers
265 views

Why continuum hypothesis implies the unique hyperreal system, ${}^{\ast}{\Bbb R}$?

On page 33, Robert Goldblatt, Lectures on Hyperreals(1998): Now it has been shown under certain set-theoretic assumption called continuum hypothesis the choice of $\mathcal F$ is irrelevant: All ...
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1answer
51 views

Action of automorphisms on Nonforking Extensions

Let $T$ be a totally transcedental theory and $M$ a model of $T$ with $A \subset M$. Let $p(x) \in S(A)$ (a complete type with parameters in $A$) and let $q(x)$ be a nonforking extension of $p(x)$ ...
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392 views

What is “induction on complexity of formula”

The fundamental theorem of ultraproducts Given a language $L$ and a family of models $M_i$ of $L$ and ultrafilter $\mathcal U$ on $I$ and $\varphi$ a formula of $L$ then $$ \prod_{i \in ...
2
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1answer
63 views

Constants in ultraproduct are well-defined

When defining ultraproduct it is defined to be product of domains of models $A_\xi$ modulus the equivalence relation by ultrafilter on index set. The relation on the product are defined by ...
2
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1answer
54 views

Elementary theory of an algebraic structures

Could someone elaborate me what the sentence "The elementary theory of finite fields is decidable" means? I'm not sure that for example if I take $x\in \mathbb{F}_4$ and $y\in \mathbb{F}_5$ then can I ...
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64 views

Finite ultraproduct

I stucked when trying to prove: If $A_\xi$ are domains of models of first order language and $|A_\xi|\le n$ for $n \in \omega$ for all $\xi$ in index set $X$ and $\mathcal U$ is ultrafilter of $X$ ...
2
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1answer
62 views

About Ehrenfeucht's theorem proof

I am re-reading the proof of Ehrenfeucht's theorem on page 90 of Mathematical Logic of Shoenfield. I have the following problem. For the sake of clarity I have highlighted in red the problematic part ...
6
votes
3answers
154 views

There is concept of finite sets that can have only one “interpretation”?

In our mind we have a naive idea of what a set is, and in nature we can only observe something that behave like a finite set, ZFC (or set theories in general) tries to catch these properties in ...
7
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1answer
100 views

What's more robust than a structural homomorphisms?

This question isn't category theory; but, category theoreticians tend to be interested in mathematical structure, so I thought the answer might exist within that knowledge base. Given two ...
4
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1answer
108 views

What is the formulation of the Least Upper Bound propierty in First Order Logic?

I've been readining about the completeness Godel's theorems. Accordingly, the axioms of $R$ in first order logic make up one of these sets that is complete and consistent. But I've always seen the ...
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2answers
333 views

Is Euclidean Geometry complete and unique

Please help me understand this concept of completeness of geometry and set me on the right path. This is my context: From wikipedia, a formal system is complete if every tautology is also a theorem. ...
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243 views

Number of automorphisms of saturated models

I have the following assignment question: Let $M$ be an $L$-model of cardinality $\kappa$. Assume $M$ is saturated. How can you show that $|\text{Aut}(M)|=2^{|M|}$? I see two possible ...
3
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1answer
61 views

Trying to define an abstract notion of a function that turns sets of sentences into the set of models satisfying those sentences.

Let $\mathsf{Sen}$ denote a Boolean algebra, thought of as a collection of sentences, and let $\mathsf{Mod}$ denote a set without any additional structure, thought of as a collection of models. I'm ...
3
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1answer
119 views

Model theory question with finiteness

It should be pretty elementary, but I can't really see it.. Cheers to anyone who can help me Let $\Psi$ be a transitive set that is a model of $ZF$. Then, $a$ is finite if and only if $\Psi \models ...
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2answers
77 views

How to show $\chi_{{}^{*}P} ={}^{*}\chi_{P}$ by transfer principle?

Let $\mathfrak{R}$ be the real number system, $(\mathbb{R},+,\cdot,<)$ and ${}^{*}\mathfrak{R}$ be the hyperreal number system $({}^{*}\mathbb{R},{}^{*}+,{}^{*}\cdot,{}^{*}<)$. Transfer ...
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4answers
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Is $\mathbb{N}$ impossible to pin down?

I don't know if this is appropriate for math.stackexchange, or whether philosophy.stackexchange would have been a better bet, but I'll post it here because the content is somewhat technical. In ZFC, ...
3
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1answer
178 views

Why every member of ${}^{*}\mathbb{R}$ is infinitely close to some member of ${}^{*}\mathbb{Q}$

Let $\mathbb{Q}$ be the set of rational numbers. Show that every member of ${}^{*}\mathbb{R}$ is infinitely close to some member of ${}^{*}\mathbb{Q}$. This is an exercise on page 180, A ...
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Is a “model” only a proper model if everything in it's definition is also explicitly constructed?

Say you have some collection of axioms and you find a set $X$ fulfilling them. And the definition of the set $X$ involves another concept, e.g. the real number, but only in a way which refers to the ...
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1answer
83 views

construction set of natural number logic

I identify the natural number $0$ with the empty set $\emptyset$, $1$ with $S(0)$, $2$ with $S(1)$, etc, etc. The axiom of infinity says $\exists x (\emptyset\in x\wedge \forall z\in x\space ...
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1answer
128 views

Inherited topology of logical Stone's spaces.

I'm asking here if the following construction is of any interest. I can not find any reference for that kind of thing, so either the subject is completely trivial, either I just don't have the correct ...
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votes
1answer
203 views

Godel number and expressibility [duplicate]

how to show that these properties of strings of symbols are expressible: 1) being a term, 2) being a formula 3) being a sentence 4) being a proof in PA and where a property (i.e., predicate) P of ...
5
votes
1answer
113 views

Which logic is stronger? SOL or $\frak{L}_{\infty,\infty}$?

Both second-order logic($\mathsf{SOL}$) and infinitary first-order logic $\frak{L}_{\infty,\infty}$ are proper extensions of first-order logic($\mathsf{FOL}$), that is, are extensions of a ...
3
votes
2answers
128 views

What counts as a standard model of arithmetic?

In my research so far, I've found that the canonical standard model of arithmetic is $\mathbb{N}$ under the addition and multiplication operations. However, I've been unable to find much on any other ...
5
votes
0answers
107 views

Model theory in terms of type spaces/Lindenbaum algebras

Are there any good references that go into some detail of known 'translations' between properties of the type space of a model and the model theoretic properties of the model? All I seem to find are ...
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4answers
110 views

A structure elementarily equivalent to $(\mathbb{N},0,\operatorname{S},<,+,\cdot)$

Given $\mathfrak{R} = (\mathbb{N},0,\operatorname{S},<,+,\cdot)$, Let $$\Sigma = \{ 0 < c, \operatorname{S}{0} < c, \operatorname{S}\operatorname{S}{0} < c, \ldots\}$$ By compactness ...
0
votes
2answers
90 views

How to show that the property of being algebraically closed is reflected by elementary extensions?

May I ask how to show that the property of being algebraically closed is reflected by elementary extensions? The reason that I want to show that is to prove the following: Prove: If ...
6
votes
2answers
197 views

The Axiom of Choice and definability

I've seen a lot of relations between the notion of the existence of a definable set with a given property and the necessity of AC is proving that there is a set with the property. For example: Under ...
5
votes
1answer
73 views

$T\vDash\psi$ equivalences

$T\vDash\psi$ means $T$ satifies $\psi$ from Tarski's definition of truth, it simply means that the sentence $\psi$ is valid in $M$. I call a sentence $\psi $ universally if it is valid in every ...
5
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1answer
124 views

Semi-formal language - Universe has at least three elements

First of all I would like to construct a semi formal sentence, such that the universum has at least three elements. My attempt: $$\exists x\exists y\exists z (x\not=y\wedge y\not=z\wedge x\not=z)$$ ...
4
votes
0answers
191 views

Difference between elementary submodel and elementary substructure

This is a really "elementary" question, forgive the pun. What is the difference between an elementary submodel and an elementary substructure (in first-order Logic)? Sincere thanks for help.
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1answer
181 views

Expressibility and numbering

A predicate $P$ is expressible (in PA) if there exists a formula $\phi(x_1,\ldots, x_n)$ of $L_A$ such that for all $m_1,\ldots, m_n$ elements of $\mathbb{N}$, we have that $P(m_1,\ldots, m_n)$ holds ...
1
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1answer
171 views

Second incompleteness and Model theorey

If we let $T$ be a consistent theory in the language of arithmetic $\mathcal{L}_A$ theory extending Peano Arithmetic — with specified numbering of formulas $\left[\cdot\right]$ and suppose that ...
3
votes
1answer
237 views

Prove that two structures are elementarily equivalent (Logic)

in an exercise I was asked to prove that (a) The structures $(\mathbb{R}^+,1,\cdot)$ and $(\mathbb{R},0,+)$ are elementarily equivalent. (b) The two structures $(\mathbb{N},<)$ and ...
3
votes
1answer
93 views

Classification of models

Let $L=\{P_0,P_1,P_2\}$ be a first order language, and let $$T=\bigg\{\Big(\forall x\ P_i(x)\Big)\vee \Big(\forall x\ \neg P_i(x)\Big):i \in \{0,1,2\}\bigg\}\\ ...
4
votes
1answer
215 views

Exclude operation symbols in signature

We probably know what a signature is, it contains a set $\sigma_{op}$ (the operation symbols), $\sigma_{rel}$ (the relation symbols) and a function $ar:\sigma_{op}\cup\sigma_{rel}\rightarrow\mathbb ...
4
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2answers
66 views

Semi-formal language

I want use semi-formal language to describe the following four points. (1) The group axioms with signature $\{*\}$ (2) The property "linear order" with signature $\{<\}$ The following properties ...
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1answer
52 views

Order isomorphism help

Let $A=B=(0,1]$ and let $\{a_i\}_{i\in\mathbb{N}}\subset A$ and $\{b_i\}_{i\in\mathbb{N}}\subset B$ are be two sequences and let $a_i\le a_j$ iff $b_i\le b_j$. is there any order preserving embedding ...
3
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1answer
142 views

Dense linear orderings isomorphism [duplicate]

First of all the definition: A dense linear order is a $\{<\}$-structure in which the following formulas are valid: Question: How can I prove that any two countable DLO's (dense linear order) ...
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2answers
40 views

cardinality of elements in a “semiring minus multiplicative identity”

In a theory that has all axioms of semiring except multiplicative identity axiom, will there be a model of the theory that has infinite elements? The model must violate multiplicative identity axiom.
3
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0answers
117 views

Does ZFC allow self-reference? [closed]

I heard that, "ZFC theory doesn't allow self-reference." But I don't know exactly what it means. As we can see in the proof of Godel's incompleteness theorem, we can use method of "Diagonalization" ...
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2answers
83 views

how do we prove that ring of characteristic $p$ has arbitrarily large models?

As title says, how do we prove that the theory that describes ring of characteristic $p$ has arbitrarily large model? I am asking for a model-theoretic approach.
0
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1answer
153 views

Gödel's Lemma and Logic with $\Delta_0$ formulas [closed]

How to prove that: There is a $\Delta_0$ formula $\theta(x,y,z)$ s.t. Naturals models for all $x$,$y$ a unique $z*\theta(x,y,z)$ and for all $k$ an element of the naturals, $z_0$,...$z_{(k-1)}$ are ...
0
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2answers
96 views

understanding provability and more about Löb's theorem

This question is an additional question for my previous question,one week ago. Link : understanding provability Fortunately, some persons kindly commented for my question. However, I think I still ...