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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Defining ultraproduct
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 ...
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1answer
29 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 ...
2
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2answers
29 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
41 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 ...
5
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3answers
106 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
72 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
44 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
50 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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Elementarily equivalent
Let $L$ be a language containing only relation symbols, and let $A$ and $B$ be
two partially isomorphic $L$-structures. Show that $A$ and $B$ are then also elementarily
equivalent, i.e. each ...
8
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91 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
48 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
104 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
71 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 ...
20
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4answers
376 views
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
121 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
60 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
45 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
54 views
Axiom schema of specification - Existence of intersection and set difference
I want to prove existence of intersection $x\cap y=\{z\in x| z\in y\}$ and set difference $x\setminus y=\{z\in x| \neg z\in y\}$using an axiom schema of specification.
My first thought was to use ...
5
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1answer
54 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
168 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 ...
4
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1answer
72 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
80 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
74 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
73 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 ...
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2answers
70 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 ...
5
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2answers
126 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
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1answer
55 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
62 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:
$$\exists x\exists y\exists z (x\not=y\wedge y\not=z\wedge x\not=z)$$
...
3
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0answers
63 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
131 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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1answer
151 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
62 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
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1answer
60 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
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1answer
116 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
46 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 ...
1
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1answer
48 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
46 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) ...
0
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2answers
26 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.
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0answers
156 views
binary string delta zero case
How to show that the binary string representing z is equal to the concatenation of the binary strings representing x and y (in that order), is a delta-zero condition?
For delta-zero, there must be ...
3
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0answers
89 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
72 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
128 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
82 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 ...
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2answers
92 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 $(G,*),$ ...
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120 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 ...
3
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1answer
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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
61 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
106 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
103 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
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3answers
156 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 ...
