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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232 views
$2\mathbb Z$ is not a definable set in the structure $(\mathbb Z, 0, S,<)$
This is (a translation of) an excerpt from a model theory textbook that shows that $2\mathbb Z$ is not a definable set in the structure $(\mathbb Z, 0, S, <)$, where $S$ is the successor function.
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2
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2answers
115 views
A theory with exactly $n$ countable models, for each $n>1$
For each $n>1$ we shall construct a first-order theory $T_n$ with exactly n countable models.
Let $n>1$, consider the language $L_n=\left\{{R,c_1,...,c_n}\right\}$, where $R$ is a binary ...
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1answer
86 views
Problem with Morley's Theorem
Greets.
Morley's theorem states that a theory which is categorical for an uncountable cardinal is categorical in all uncountable cardinals.
My problem with the theorem is that I haven't found a ...
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2answers
111 views
Finite Set of Models
This is only directed towards logicians, model theorists etc.. I am reading "Model Theory" by Keisler and Chang and have encountered the following question.
Let $\Psi = \{M_1,...,M_n \}$ be a finite ...
3
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2answers
82 views
First order sentence true in $\mathbb{Q}$ but not in $\mathbb{R}$.
I have the following assignment question:
Find a first order sentence that is true in $\langle\mathbb{Q},+,\cdot ,0,1\rangle$ but not in $\langle \mathbb{R},+,\cdot,0,1\rangle$.
Most of what I ...
1
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2answers
94 views
Boolean algebra spectrum
The first-order spectrum of a theory, is the set of cardinalities of its finite models. Finite models of Boolean algebras are informally n-dimensional cubes, therefore boolean algebra spectrum is the ...
3
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2answers
126 views
Does Löwenheim-Skolem fail for $\mathcal{L}_{\omega_1\omega}$?
Does descendant or ascendant Löwenheim-Skolem fail for $\mathcal{L}_{\omega_1\omega}$ -logic?
2
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2answers
144 views
statement that is consistent in ZFC but the negation of it can be both consistent and inconsistent in ZFC or vice versa
Is there any known case where ZFC system is known to be consistent with a statement, but is also consistent with the negation of the statement? Or vice versa.
Also, when we say ZFC is consistent with ...
1
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2answers
94 views
Existential quantification as projection
I read the following statement here regarding equivalence of existential quantification and projection of basic relations in model theory.
The operation of taking image under a coordinate ...
4
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2answers
105 views
Gentle introduction into stability and classification theory
I am badly looking for a (very) gentle introduction into stability and classification theory answering at least some of the following questions:
Why is a stable theory called "stable"?
What is a ...
6
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1answer
129 views
Łoś's Theorem holds for positive sentences at reduced products in general?
Let $ \mathcal{L} $ be a language for first-order logic whose logical primitives are $ \neg$, $\vee$, $\wedge$, $\forall$, and $\exists$, with the usual formation rules. A sentence $ \sigma $ is ...
1
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1answer
148 views
Equivalence to a universal formula
I am trying to prove the equivalence of the following assertions (Exercise 2.5.12 from Marker "Model Theory: An Introduction").
There is a universal formula $\psi(\bar v)$ such that $T \models ...
3
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0answers
91 views
Some intuition behind o-minimal systems.
I am following the definitions of an o-minimal system as defined in "Tame Topology and O-minimal Structures" by Lou van den Dries.
It is immediate from the definition that the graph of $\sin(x)$ is ...
8
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1answer
166 views
Do ordered fields and archimedian ordered fields have the same first-order theory?
Let us suppose that the first-order language of ordered fields has symbols for addition, subtraction, multiplication and order, and constant symbols for 0 and 1.
An ordered field is said to be ...
3
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0answers
69 views
An application of Descriptive set theory in Model theory.
In page 162 of D.Marker Model theory book he proved that the set $S_n(F,T)$ of all $F$-types realized by some $n$-tuple in some countable model of $T$ is analytic (this is with any $F$:= $countable$ ...
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1answer
65 views
Question about a defined function in D.Marker Model Theory book.
In page 162 of D.Marker Model theory book he proved that the set $S_n(F,T)$ of all $F$-types realized by some $n$-tuple in some countable model of $T$ is analytic (this is with any $F$:= $countable$ ...
3
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1answer
170 views
How to exhibit models of set theory
Even though $\mathbb{N}$ cannot be defined by first order means, it can be defined by second order means. Anyway: it can be defined, and there is no doubt, which abstract structure $\mathbb{N}$ ...
4
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2answers
114 views
Nonstandard extension of a function with a limit
Question 1. Let $g : \mathbb{R} \to \mathbb{C}$ with $g(y) = \lim_{x \to \infty} f(x,y)$, where $f : \mathbb{R}^2 \to \mathbb{C}$. Is it correct that the nonstandard extension $^*g$ will have $x \in ...
2
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1answer
84 views
On the Decision Problem for Two-variable First-Order Logic
I have a question concerning the model construction of the $\forall \forall \land \forall \exists$
- Scott sentence on page 6 in this paper:
www.cs.rice.edu/~vardi/papers/basl96.ps.gz
Why do we ...
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0answers
46 views
Morphisms of Euclidean Geometry
In Euclidean geometry $\mathbb{E}^2(\cong \mathbb{C})$, the group $G=\{z\mapsto raz+b, z\mapsto ra\bar{z}+b\colon a,b\in \mathbb{C},|a|=1, r\in \mathbb{R}^{+} \}$ is precisely the group of ...
6
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1answer
204 views
Is there a difference between a model and a representation?
I'm thinking of models in logic here, vs. e.g. group representations.
Is there a difference between a model and a representation?
Could one not explain both at the same time?
A model gives an ...
4
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0answers
82 views
is a group that eliminates quantifiers in the group language Abelian?
Let $G$ be a group. Consider it as a structure in the language of groups $L=(e,\cdot, (-)^{-1})$. Suppose that $G$ eliminates quantifiers in this language. Is it true that $G$ must be Abelian then?
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3answers
180 views
How are the full semantics of SOL and HOL specified?
In relation to this question about the "fundamental" character of possible logical systems, I realized that I just had an intuitive (and so, inadequate) understanding of the way logics higher than FOL ...
3
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1answer
98 views
question about Skolem theories
Right now I am reading a proof of Downward Löwenheim-Skolem theorem in Hodges, but I am slightly confused about a proof Hodges makes. Let me write down some of the definitions.
Definition: Let ...
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63 views
isomorphism types of countable elementary submodels
I am reading Abraham's article on proper forcing (in Handbook of Set Theory). I met problems in reading the proof of Theorem 2.10,
what is the isomorphism type of a countable elementary submodel?
...
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46 views
Density for archimidean extension of real closed field
Let $(k,<)$ be a real closed field and $L|K$ an ordered extension such that $\forall x\in L \exists y\in k\; (x<y)$.
Is $k$ dense in $L$?
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3answers
172 views
Weak categoricity in first order logic
In a certain sense, only finite structures are definable up to isomorphism in first order logic. But if we rely on a metatheory containing a sufficient strong set theory (like required for second ...
4
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1answer
89 views
the solution of ode can be encoded as first order logic
Consider the following sytem of ODEs
$\dot{x}= Ax$, and given $x(0)$,
where $A$ is a $n\times n$ matrix with rational entries.
Can I encode the solution, say $x(t)$ for a given $t$, as a first ...
2
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2answers
119 views
Am I allowed to realize one object twice within one set-theory?
Say I consider a set theory with the Axioms of Extensionality and the Axiom of Pairing.
As I understand it, stating the axiom allows me to make a definition like
$$(a,b):=\{\{a\},\{a,b\}\}$$
and ...
3
votes
2answers
185 views
Why is completeness theorem true?
Who can teach me completeness theorem? Thanks! Recommending a book is also welcome.
More specifically, it says that if a statement is true in all models of a theory, then it has a proof from this ...
2
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1answer
107 views
first order logic question model
suppose we have a model for a language in first order logic $ M=<D,I> $
such that D is the domain and I is the interpetation such that for every $ a \in D $ we have a closed noun (a noun with no ...
2
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1answer
88 views
Absoluteness and categories
From the wikipedia article on the Skolem paradox:
A central goal of early research into set theory was to find a first order axiomatisation for set theory which was categorical, meaning that the ...
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1answer
96 views
Saturated Boolean algebras in terms of model theory and in terms of partitions
Let $\kappa$ be an infinite cardinal. A Boolean algebra $\mathbb{B}$ is said to be $\kappa$-saturated if there is no partition (i.e., collection of elements of $\mathbb{B}$ whose pairwise meet is $0$ ...
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2answers
74 views
Logic question proving something about compactness
Let $\Sigma$ be a set of formulas. There's a finite set $\Lambda \subseteq \Sigma$.
I'm asked to prove or disprove that $\Sigma$ has a model if and only if $\Lambda$ has a model.
It seems to me ...
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8answers
380 views
Learning Model Theory
What books/notes should one read to learn model theory? As I do not have much background in logic it would be ideal if such a reference does not assume much background in logic. Also, as I am ...
2
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2answers
56 views
Embedding of standard model of arithmetic to PA-model
I am working on the following problem:
Let $ S_{Arithmetic} = \{+, *, 0, 1\}, \mathfrak{M} $ a model for PA (first-order peano axioms) }, and $ \mathbb{N} = (\mathbb{N},+ ^{\mathbb{N}}, ...
4
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1answer
137 views
Lindenbaum algebra is a free algebra
The following is a continuation of this question.
I would like to prove that the Lindenbaum algebra is a free algebra. Hopefully I would like to hear hints on how to proceed in the 'right' ...
4
votes
1answer
100 views
Are there simple counterexamples to a strengthening of omitting types theorem
The famous Ehrenfeucht's omitting types theorem states that for any countable set of nonisolated types (without parameters), there is a (countable) model such that it does not realize any of them.
A ...
3
votes
1answer
165 views
Complete/incomplete theory
I am thinking about completeness and incompleteness of theory's, and to illustrate both properties i am thinking of how to build an complete system, and then turn it into an incomplete one.
Example. ...
2
votes
1answer
98 views
Is there a relationship between the Compactness Theorem and the upward Lowenheim-Skolem Theorem in FOL?
Is there a relationship between the Compactness Theorem and the upward Lowenheim-Skolem Theorem in FOL?
I was thinking of another post of mine "Why accept the axiom of infinity?" when I though, ...
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0answers
109 views
Can we find a nice definition of Congruence in Topology?
According to my knowledge, quotient structure is a original structure divided by a congruence. However, quotient topology space is defined this way.
Quotient_topology
In this way, $\sim$ is only said ...
0
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1answer
69 views
Relations of language/theory/signature
Say that the language of the first order logic is the collection of symbols that can be used in the formulas + the grammar (the rules that specify how they can be combined)?
1) However, the signature ...
2
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1answer
120 views
Explanation of how models can differ on $\omega$?
Assuming set theory (here, ZF) is consistent, there is a model $V$ of ZF, the universe of all sets.
So, there is a $\omega^V\in V$. A set $A\in V$ is countable iff a bijection $f\in V$ exists ...
3
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2answers
102 views
dense linear orders DLO
I am asked to prove that
if I have two models of dense linear orders DLOs WITH the minimum and maximum. must be izomorpic to each other by fining direct izomorphy.
I seem to always get stuck ...
6
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2answers
117 views
What's the name of this operator?
Let $f,g$ be functions in $C^A$ and $C^B$ respectively.
Let $\boxtimes:C^A \times C^B \to (C\times C)^{A \times B}$ s.t.
$f\boxtimes g(a,b)=(f(a),g(b))$
It seems not the tensor product, nor ...
1
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1answer
58 views
How to formally describe this Uppaal automata?
I have the following simple automata:
What I'm looking for is a formal description of this based on the definition here
$A=(\Sigma,\Gamma,S,s_0,\delta,\omega, F)$
How to declare all the ...
0
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1answer
166 views
Models of propositional logic
Define a theory of propositional calculus as the set $T$ of axioms (expressed in propositional calculus) and a set of valid symbols.
What I would like to see are some examples of theories in ...
2
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2answers
94 views
Does Bernstein theorem hold for models with elementary functions?
Bernstein theorem is a general pattern that occurs in many areas of mathematics (see the Wikipedia article for some examples). Does it hold for arbitrary models with elementary embeddings?
To be more ...
0
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1answer
60 views
Factor in ultraproduct
The general method for getting ultraproducts uses an index set I, a structure $M_i$ for each element i of I (all of the same signature), and an ultrafilter U on I. The usual choice is for I to be ...
2
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2answers
193 views
confusion regarding compactness theorem
I am getting somehow confused of compactness theorem.
The compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This ...
