Concept
Model theory
Overview
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). 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. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other.
As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954.
Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory.
Compared to other areas of mathematical logic such as proof theory, model theory is often less concerned with formal rigour and closer in spirit to classical mathematics.
This has prompted the comment that "if proof theory is about the sacred, then model theory is about the profane".
The applications of model theory to algebraic and Diophantine geometry reflect this proximity to classical mathematics, as they often involve an integration of algebraic and model-theoretic results and techniques. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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190. The difficulty that M. Bayle has imagined in connexion with Strato seems a little too subtle and far-fetched. That is termed: _timere, ubi non est timor_. He makes another difficulty, which has just as slight a foundation, namely, that God would be subjected to a kind of _fatum_. Here are his words (p.
Gottfried Wilhelm Leibniz, Theodicy Essays on the Goodness of God, the Freedom of Man and the Origin of Evil
States what Model theory is; the author is tied to Model theory in the knowledge graph.
Any such select set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates.
Alfred North Whitehead, Science and the modern world
States what Model theory is.
Geometry passed through a stage of abstraction like that examined in connection with arithmetic. Beginning with the discovery of non-Euclidian geometry, it has been becoming more and more evident that a line need not be a name for an aspect of a physical object such as the ridge-pole line of a house and the like,…
John Dewey, Creative Intelligence
States what Model theory is.
Philosophers who discussed Model theory
- Gottfried Wilhelm Leibniz
- Willard Van Orman Quine
- Rudolf Carnap
- Alfred Tarski
- Hilary Putnam
- Kazimierz Ajdukiewicz
- Hans Kamp
- Richard Milton Martin
- Newton da Costa
- John Lane Bell
- Ian Rumfitt
Related ideas
- Formal language
- First-order logic
- Independence (mathematical logic)
- Gödel's completeness theorem
- Skolem's paradox
- Springer Science+Business Media
- Lwów–Warsaw school
- Interpretation (logic)
- Forcing (mathematics)
- Compactness theorem
- Zermelo–Fraenkel set theory
- Kurt Gödel
- Proof theory
- Mathematical logic
- Sentence (mathematical logic)
- Boston
- Set theory
- Topological space
- University of California, Berkeley
- Löwenheim–Skolem theorem
- Logical consequence
- Boolean algebra (structure)
- Formal system
- Structure (mathematical logic)
Opposing perspectives
Place in the tradition
Part of Analytic philosophy.