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 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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35 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
50 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 ...
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1answer
49 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
162 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 ...
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68 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 ...
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2answers
77 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 ...
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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
64 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
68 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 ...
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2answers
115 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 ...
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1answer
54 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
60 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
47 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
130 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 ...
