Branches of Mathematics
Model Theory
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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.
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 Cross-Tradition Connections
Sources
1. Wikipedia: Model Theory
Wikimedia FoundationLead sectionQuote, Lead section
model theory is the study of the relationship between formal theories and their models
1. Wikipedia: Model Theory
Wikimedia FoundationCompactness and the Lowenheim-Skolem theorem sectionQuote, Compactness and the Lowenheim-Skolem theorem section
every infinite structure in a countable signature has a countable elementary substructure
1. Wikipedia: Model Theory
Wikimedia FoundationFirst-order logic sectionQuote, 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
Stone-Weierstrass Theorem (Wikipedia)
Wikipedialead paragraphQuote, 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).
Stone-Weierstrass Theorem (Wikipedia)
Wikipedialead sectionQuote, lead section
The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself.
Stone-Weierstrass Theorem (Wikipedia)
WikipediaStatement sectionQuote, Statement section
The completeness theorem makes a close link between model theory, which deals with what is true in different models, and proof theory, which studies what can be formally proven in particular formal systems.
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