Questions tagged [model-theory]

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.

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Is there a 0-1 law for the theory of groups?

For each first order sentence $\phi$ in the language of groups, define : $$p_N(\phi)=\frac{\text{number of nonisomorphic groups $G$ of order} \le N\text{ such that } \phi \text{ is valid in } G}{\text{number of nonisomorphic groups of order} \le…
Dominik
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Complete, Finitely Axiomatizable, Theory with 3 Countable Models

Does there exist a complete, finitely axiomatizable, first-order theory $T$ with exactly 3 countable non-isomorphic models? A few relevant comments: There is a classical example of a complete theory with exacly $3$ models. This theory is not…
Primo Petri
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How is the Gödel's Completeness Theorem not a tautology?

As a physicist trying to understand the foundations of modern mathematics (in particular Model Theory) $-$ I have a hard time coping with the border between syntax and semantics. I believe a lot would become clearer for me, if I stated what I think…
Lurco
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I don't understand Gödel's incompleteness theorem anymore

Here's the picture I have in my head of Model Theory: a theory is an axiomatic system, so it allows proving some statements that apply to all models consistent with the theory a model is a particular -- consistent! -- function that assigns every…
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In plain language, what's the difference between two things that are 'equivalent', 'equal', and 'identical'?

In plain language, what's the difference between two things that are 'equivalent', 'equal', 'identical', and isomorphic? If the answer depends on the area of mathematics, then please take the question in the context of logical systems and…
Hal
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Distinguishing non-isomorphic groups with a group-theoretic property

I am teaching a first-semester course in abstract algebra, and we are discussing group isomorphisms. In order to prove that two group are not isomorphic, I encourage the students to look for a group-theoretic property satisfied by one group but not…
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Do the axioms of set theory actually define the notion of a set?

In Henning Makholm's answer to the question, When does the set enter set theory?, he states: In axiomatic set theory, the axioms themselves are the definition of the notion of a set: A set is whatever behaves like the axioms say sets behave. This…
justin
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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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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, we can prove that the (second-order) Peano axioms…
goblin GONE
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Is $ \pi $ definable in $(\Bbb R,0,1,+,×, <,\exp) $?

Is there a first-order formula $\phi(x) $ with exactly one free variable $ x $ in the language of ordered fields together with the unary function symbol $ \exp $ such that in the standard interpretation of this language in $\Bbb R $ (where $ \exp $…
Dominik
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Murder at Hilbert's Hotel! A study of the testimony of infinitely many suspects.

I'm sorry if this is a duplicate in any way. I doubt it's an original question. Due to my ignorance, it's difficult for me to search for appropriate things. Motivation. This question is inspired by Exercise 1.2.16 of these logic notes by S. G.…
Shaun
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Does any uncountable complete theory have exactly two countable models?

The following is a theorem by Vaught. Theorem. Let $T$ be a complete theory in a countable language. Then, $T$ cannot have exactly two countably infinite models (up to isomorphism). A proof can be found at [Tent-Ziegler, A Course in Model…
ll_n
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Applications of model theory to analysis

Some of the more organic theories considered in model theory (other than set theory, which, from what I've seen, seems to be quite distinct from "mainstream" model theory) are those which arise from algebraic structures (theories of abstract groups,…
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Does there exist any uncountable group , every proper subgroup of which is countable?

Does there exist an uncountable group , every proper subgroup of which is countable ?
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FO-definability of the integers in (Q, +, <)

With $Q$ the set of rational numbers, I'm wondering: Is the predicate "Int($x$) $\equiv$ $x$ is an integer" first-order definable in $(Q, +, <)$ where there is one additional constant symbol for each element of $Q$? I know this is the case if…
Michaël Cadilhac
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