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 Connections
Associated With
Includes
Source Wikipedia: Alfred Tarski
Source Ehrenfeucht-Fraisse game, Wikipedia
Source Fraisse limit (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
View the Source 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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