Model theory is the branch of mathematical logic that studies the relationship between formal theories, collections of sentences in a formal language, and their models, the mathematical structures in which those sentences hold true. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Central QuestionHow many models a theory has, of what sizes, how those models relate to one another, and how far a theory's models are constrained or left open by its formal language alone. 1 Key DebateHow closely truth in a model and provability from axioms track each other. Godel's completeness theorem ties the two together for first-order logic, linking model theory's question of what is true in different models to proof theory's question of what can be formally proven, a link later work, including Godel's own incompleteness theorems, showed does not extend to guarantee that every true statement about a fixed structure is provable at all. 1 Classification
Pure or Applied Model Theory
Filter Results6 entries
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Associated With
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Source Wikipedia: Alfred Tarski
Source Boris Trakhtenbrot (Wikipedia)
Source Ehrenfeucht-Fraisse game, Wikipedia
Source Fraisse limit (Wikipedia)
Source Jerzy Łoś (Wikipedia)
Source Wikipedia: Joseph R. Shoenfield
Source Los Tarski preservation theorem (Wikipedia)
Source Type (model theory) (Wikipedia)
Source Wikipedia: Robert Lawson Vaught
Source Wikipedia: Simon B. Kochen
Source Trakhtenbrot's theorem (Wikipedia)
Source Vaught Conjecture (Wikipedia)
Source Vaught Conjecture (Wikipedia)
Sources
1. Wikipedia: Model Theory
Wikimedia FoundationLead section
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure) and their models (those structures in which the statements of the theory hold).
Compactness and the Lowenheim-Skolem theorem section
every infinite structure in a countable signature has a countable elementary substructure
First-order logic section
a theory has a model if and only if it is consistent, i.e. no contradiction is proved by the theory
History section
who first used the term "Theory of Models" in publication in 1954
In Group: Mathematical Logic and Proof, lead paragraph
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure) and their models (those structures in which the statements of the theory hold).
View the Source Boris Trakhtenbrot (Wikipedia)
Includes: Boris Trakhtenbrot, Biography, Trakhtenbrot theoremQuote, Includes: Boris Trakhtenbrot, Biography, Trakhtenbrot theorem
He also discovered and proved the theorem in logic, model theory, and computability theory now known as Trakhtenbrot's theorem.
Jerzy Łoś (Wikipedia)
Includes: Jerzy Łoś, ContributionsQuote, Includes: Jerzy Łoś, Contributions
He is especially known for his work in model theory
Type (model theory) (Wikipedia)
Includes: Omitting Types Theorem, Lead sentenceQuote, Includes: Omitting Types Theorem, Lead sentence
In model theory and related areas of mathematics, a type is an object that describes how a (real or possible) element or finite co
Los Tarski preservation theorem (Wikipedia)
Includes: Los-Tarski Theorem, Lead sentenceQuote, Includes: Los-Tarski Theorem, Lead sentence
substructure preservation theorem, is a result in model theory that states the first-order properties are preserved when passing t
Fraisse limit (Wikipedia)
Vaught Conjecture (Wikipedia)
Includes: Vaught Conjecture, Lead sentence
ture is a conjecture in the mathematical field of model theory originally proposed by Robert Lawson Vaught in 1961.
Includes: Vaught's Theorem, Article text
The Vaught conjecture is a conjecture in the mathematical field of model theory originally proposed by Robert Lawson Vaught in 1961.
View the SourceEhrenfeucht-Fraisse game, Wikipedia
Wikipedia: Alfred Tarski
Includes: Alfred Tarski, LeadQuote, Includes: Alfred Tarski, Lead
best known for his work on model theory
Wikipedia: Robert Lawson Vaught
Includes: Robert Lawson Vaught, LeadQuote, Includes: Robert Lawson Vaught, Lead
one of the founders of model theory
Wikipedia: Joseph R. Shoenfield
Includes: Joseph R. Shoenfield, LeadQuote, Includes: Joseph R. Shoenfield, Lead
Shoenfield worked on recursion theory, model theory and axiomatic set theory.
Wikipedia: Simon B. Kochen
Includes: Simon B. Kochen, LeadQuote, Includes: Simon B. Kochen, Lead
Simon Bernhard Kochen (; born 14 August 1934) is a Canadian mathematician, working in the fields of model theory, number theory and quantum mechanics. == Education and career == Kochen was born in Antwerp, Belgium, and escaped the Nazis with his family, thanks to a courageous Norwegian ship captain.
Trakhtenbrot's theorem (Wikipedia)
Includes: Trakhtenbrot's Theorem, Lead paragraphQuote, Includes: Trakhtenbrot's Theorem, Lead paragraph
In logic, finite model theory, and computability theory, Trakhtenbrot's theorem (due to Boris Trakhtenbrot) states that the problem of validity in first-order logic on the class of all finite models is undecidable.
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