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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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 ...
5
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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
173 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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0answers
70 views

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 ...
3
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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 ...
5
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126 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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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
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2answers
127 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
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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
109 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
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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
192 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
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0answers
188 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
180 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 ...
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vote
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
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1answer
232 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
92 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
211 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
138 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
39 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" ...
0
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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 ...
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2answers
95 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 ...
5
votes
2answers
124 views

Is a topological space a structure?

In model theory, a structure (or "model") is typically defined as a set together with some finitary relations and/or operations on that set. For instance, a group can be viewed as a pair $(G,*),$ ...
14
votes
1answer
308 views

What lessons have mathematicians drawn from the existence of non-standard models?

So, as someone whose knowledge of mathematics has always come from studying it with an eye towards philosophical/foundational issues and studying it with other philosophers (who are not primarily ...
4
votes
1answer
87 views

Question regarding inexpressibility results over finite models using compactness and the Löwenheim–Skolem theorem

In the book Elements of finite model theory by Leonid Libkin, they show that the parity query for structures over an empty vocabulary is not first order definable. They do this by constructing two ...
4
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1answer
74 views

Two homogenous structures realizing the same types are isomorphic

Let $M$ and $N$ be two countable, homogeneous structures, and assume that they both realize the same types with a finite number of variables. Does it follow that $M$ and $N$ are isomorphic? What if ...
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1answer
116 views

understanding provability

I am still confused about provability. . . Let a statement P is, sort-of-says like this. P: ( "X is provable" ∧ "P is provable" ) If ( X is provable ∧ P is provable ) is provable → (P is ...
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1answer
223 views

SHOW that there are infinitely many equivalence classes of formulas

Let $\mathcal{Q}$ denote the additive group of rational numbers, i.e. the structure $\left<\mathbb{Q}; +; 0\right>$. Let $\mathcal{L}$ be the language of $\mathcal{Q}$ and let $T$ be the ...
12
votes
3answers
223 views

Axiom of Choice and Determinacy

In my set theory course we have been talking about the axiom of determinacy. One of the first things we showed was that $AD$ and $AC$ are incompatible. We later showed that $ZF+AD$ implies the ...
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vote
1answer
89 views

Spectrum and sentences

Is there an example of a language with only one spectrum equal to the even numbers? Also, is there an example of a language with only relation which has spectrum equal to the set of non-primes? A ...
3
votes
2answers
126 views

Standard model and formulas in logic

If $\phi(v)$ is a formula defining the prime numbers in the standard model, then are there nonstandard elements satisfying $\phi(v)$? What if $\phi(v)$ is a formula for powers of $2$? Also, what if ...
3
votes
2answers
95 views

Non-Archimedan Groups

I'm trying to think of an explicit example of a non-Archimedian group. The definition of Archimiedean is s.t. if for all $x$ and $y$, there is some $m$ a Natural number s.t. $mx = \underbrace {x + x ...
0
votes
1answer
55 views

Model THeories not equivalent

How can we show that Th(Rationals) does not equal Th(Integers) does not equal Th(Naturals)? Where Th(M) is a the set of all L sentences which are true in a model, M. I know that L is defined as = ...
4
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0answers
165 views

What is the correspondence between the semantic compactness of logics and the compactness of Stone spaces?

According to the answer by François G. Dorais, we know that a logic $\mathfrak L$ is compact iff its Stone space of the Lindenbaum–Tarski algebra of the empty theory (w.r.t the deductive system) is ...
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1answer
45 views

Is there a property that isolates the formulas that remain valid as you move from the natural numbers to the integers?

Is there a property that isolates the formulas that remain valid (or whose translations into the expanded language remain valid) as you move from the theory of the natural numbers to the theory of the ...
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1answer
82 views

What is the difference between First-Order Structures and Kripke Structures?

In the SEP article on Model Theory by Wilfrid Hodges (here), he writes: Particular kinds of model theory use particular kinds of structure; for example mathematical model theory tends to use ...
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2answers
106 views

Applying the compactness theorem

Using a Hilbert system: L is a FOL (First order language) with R, where R is a single binary predicate symbol. Suppse A = ⟨V,E⟩ is a structure for this language domain V = |A|. Suppose also that E = ...
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2answers
79 views

Non-isomorphic structures with equal cardinality

Let $\mathfrak{A}=(\mathbb{N},S,0)$ be a structure where $S$ is the sucessor function. Let $\mathfrak{B} =(\mathbb{N}\times \{0\} \cup \mathbb{Z} \times\{1\} ,S, 0)$ with $0 = (0,0)$ and $$ S(k,i) ...
13
votes
2answers
290 views

A sentence false in a field of characteristic $0$ but true in all fields of positive characteristic?

Consider the language $L=\{+,\cdot, 0, 1\}$ of rings. It is easy to show using compactness that if $\sigma$ is a sentence that holds in all fields of characteristic $0$, there is some $N\in \mathbb N$ ...
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votes
1answer
56 views

Specturm 3b in a link

Can someone help with #3b in this link: http://homepages.math.uic.edu/~marker/math502f09/ps3.pdf I am trying to practice idea of spectrum, but cannot quite understand whatt they are asking for. ...
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3answers
241 views

Is Induction Independent of the Other Axioms of PA?

I am trying to come up with a model of first order Peano Arithmetic (PA) where induction fails. Let $PA^{-IND}$ have the same axioms as PA except the first order induction axiom schema is replaced ...