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 languages. Not to be confused with mathematical modeling.
Questions tagged [model-theory]
4344 questions
37
votes
6 answers
What is an efficient nesting of mathematical theorems?
Various mathematical areas of research evolved from a wide and diverse range of questions. Many are physical in nature or come from informatics/computer science, some are procedural or optimization problems and so on. Often patterns emerge and lead…
Nikolaj-K
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19
votes
7 answers
Does a finite first-order theory which has a model always have a finite model?
I'm curious whether this is true or not:
Let T be a finite, first-order theory. If T has a model, then T has a finite model.
I would assume the answer is 'yes', but I wanted to make sure I haven't missed something obvious. The reason I believe…
Koz Ross
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13
votes
1 answer
What's the motivation behind saturated models?
In Model Theory by Chang & Keisler, saturated models are introduced on page 100.
A model $\mathfrak A$ is said to be $\omega$-saturated iff for every finite set $Y \subset A$, every set of formulas $\Gamma(x)$ of $\mathcal{L}_Y$ consistent with…
Nate
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11
votes
2 answers
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 ideas/connections/intuitions here:
Definable sets. Since $M$…
FPP
- 2,103
11
votes
1 answer
If a theory has a countable $\omega$-saturated model does it need to have only countable many countable models?
If a theory has countably many countable models (up to isomorphism) then it has at countably many types, and it follows that there exists a countable $\omega$-saturated model of such theory.
If a theory has a countable $\omega$-saturated model then…
Santiago C.
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11
votes
2 answers
counterexample to the omitting types in uncountable language
Could anyone remind me of an example of an uncountable non-isolated complete type that cannot be omitted?
Primo Petri
- 5,174
11
votes
2 answers
Why does creating a model show consistency?
As per the title, why does the ability to generate a model from axioms prove they are consistent?
Joey
- 111
9
votes
2 answers
Definability in a given structure
I want to prove the following statements:
Is the function "sin" definable in the structure $(\Bbb{R},<,+,\cdot,0,1)$, that is does there exists a formula $\phi=\phi(x_0,x_1)$ such that for all $a,b\in\Bbb{R}$: $sin(a)=b$ iff…
Prime
- 91
- 1
8
votes
1 answer
Model Theory (Hodges), Section 2.1, Exercise 13
Let $K$ be a field of characteristic 0, $n$ a positive integer, and $G$ the group $GL_n(K)$ of invertible linear transformations on $K^n$. Show that the following subsets of $G$ are $\emptyset$-definable: (a) the set of all scalar matrices; (b) the…
Adam Gutter
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8
votes
1 answer
Are infinite subsets of the real field definable by a single formula?
Consider the structure $(\mathbb{R},+,*,0,1,<)$. We adjoin to it a subset $S$ of $\mathbb{R}$. Is it possible to give a single formula $F$ in the expanded language such that $F$ is true precisely when $S$ is an infinite subset of the reals? I know…
user107952
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8
votes
0 answers
Can any o-minimal structure be expanded to an o-minimal structure with definable choice?
Let $\mathcal{M}=(M,<,\ldots)$ be an o-minimal structure, namely a linearly ordered (by $<$) first order structure such that every definable set in $M$ is finite union of points and intervals $(a,b)$ where $a\in M\cup\{-\infty\}$ and $b=M\cup…
Anguepa
- 3,129
8
votes
1 answer
Are there any non-trivially 'potentially categorical' first order theories?
First some background: Two structures $\mathcal{M}$ and $\mathcal{N}$ are potentially isomorphic if there is a non-empty family of finite partial isomorphisms between them such that
for any member of the family $f$ and $x\in\mathcal{M}$, there is…
James Hanson
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7
votes
0 answers
Is there a recursively axiomatizable theory which has no independent recursive axiomatization?
Does there exist a theory $T$ (in the sense of model theory) such that $T$ is recursively axiomatizable, but there is no independent recursive axiomatization of $T$? Or, does every recursively axiomatizable theory have an independent recursive…
user107952
- 20,508
7
votes
1 answer
Definability of algebraic numbers in the real field
Consider the structure $(\mathbb{R},+,-,*,0,1,\leq)$. We adjoin to it a constant $r$. Is there a set $S$ of formulas in that expanded language, perhaps an infinite set, such that the members of $S$ are jointly satisfied iff $r$ is an algebraic real…
user107952
- 20,508
7
votes
1 answer
Theory of (Z,+) has uncountably many 1-types
I'm working on some exercises in model theory, but on this one I don't know how to start. Please help to solve this.
Prove that $\text{Th}(\mathbb{Z},+)$, the theory of the structure $(\mathbb{Z},+)$, has uncountably many $1$-types.
natural
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