Although it took 40 years to notice, the existence of these ultraproduct constructions highlights that model theory has a role to play in the subject. There are many interesting current directions to pursue but we wish to concentrate on three general themes:. There are several things that model theory brings to the table in this endeavour. First of all, the theory of an algebra is an invariant which is complementary to many of the operator algebraic invariants on offer.
The utility and consequences of recognizing when two algebras do not have the same theory will be highlighted below. Second, model theory provides methods of constructing examples which are different from those in operator algebra. The primary example is model theoretic forcing which plays a prominent role in [thebook]. Although to date the examples constructed have been modest, refocusing attention on the construction of specific examples with this technique in mind could pay dividends. Model theory can be used for both clarifying concepts and identifying good questions to ask.
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Michael Morley 's highly non-trivial result that for countable languages there is only one notion of uncountable categoricity was the starting point for modern model theory, and in particular classification theory and stability theory:. Uncountably categorical i. Set theory which is expressed in a countable language , if it is consistent, has a countable model; this is known as Skolem's paradox , since there are sentences in set theory which postulate the existence of uncountable sets and yet these sentences are true in our countable model.
Particularly the proof of the independence of the continuum hypothesis requires considering sets in models which appear to be uncountable when viewed from within the model, but are countable to someone outside the model.
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In the other direction, model theory itself can be formalized within ZFC set theory. The development of the fundamentals of model theory such as the compactness theorem rely on the axiom of choice, or more exactly the Boolean prime ideal theorem. Other results in model theory depend on set-theoretic axioms beyond the standard ZFC framework.
For example, if the Continuum Hypothesis holds then every countable model has an ultrapower which is saturated in its own cardinality. Similarly, if the Generalized Continuum Hypothesis holds then every model has a saturated elementary extension. Neither of these results are provable in ZFC alone. Finally, some questions arising from model theory such as compactness for infinitary logics have been shown to be equivalent to large cardinal axioms. A field or a vector space can be regarded as a commutative group by simply ignoring some of its structure.
The corresponding notion in model theory is that of a reduct of a structure to a subset of the original signature. The opposite relation is called an expansion - e.
The terms reduct and expansion are sometimes applied to this relation as well. Given a mathematical structure, there are very often associated structures which can be constructed as a quotient of part of the original structure via an equivalence relation. An important example is a quotient group of a group. One might say that to understand the full structure one must understand these quotients.
When the equivalence relation is definable, we can give the previous sentence a precise meaning. We say that these structures are interpretable. A key fact is that one can translate sentences from the language of the interpreted structures to the language of the original structure.
Thus one can show that if a structure M interprets another whose theory is undecidable , then M itself is undecidable. This is the heart of model theory as it lets us answer questions about theories by looking at models and vice versa. One should not confuse the completeness theorem with the notion of a complete theory. A complete theory is a theory that contains every sentence or its negation. Importantly, one can find a complete consistent theory extending any consistent theory.
In particular, the theory of natural numbers has no recursive complete and consistent theory. Non-recursive theories are of little practical use, since it is undecidable if a proposed axiom is indeed an axiom, making proof-checking a supertask. The compactness theorem states that a set of sentences S is satisfiable if every finite subset of S is satisfiable. In the context of proof theory the analogous statement is trivial, since every proof can have only a finite number of antecedents used in the proof.
In the context of model theory, however, this proof is somewhat more difficult. Model theory is usually concerned with first-order logic , and many important results such as the completeness and compactness theorems fail in second-order logic or other alternatives. In first-order logic all infinite cardinals look the same to a language which is countable.
By Stone's representation theorem for Boolean algebras there is a natural dual notion to this. Many important properties in model theory can be expressed with types.
Further many proofs go via constructing models with elements that contain elements with certain types and then using these elements. The theory has quantifier elimination. This allows us to show that a type is determined exactly by the polynomial equations it contains.
This is finer than the Zariski topology. Model theory as a subject has existed since approximately the middle of the 20th century. However some earlier research, especially in mathematical logic , is often regarded as being of a model-theoretical nature in retrospect.
Tarski's work included logical consequence , deductive systems , the algebra of logic, the theory of definability, and the semantic definition of truth , among other topics. His semantic methods culminated in the model theory he and a number of his Berkeley students developed in the s and '60s. These modern concepts of model theory influenced Hilbert's program and modern mathematics.
From Wikipedia, the free encyclopedia.
This article is about the mathematical discipline. For the informal notion in other parts of mathematics and science, see Mathematical model. Main article: Universal algebra. Main article: Finite model theory. Main article: First-order logic. Main article: Reduct.
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Main article: Interpretation model theory. Main article: Type model theory.
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The theorem has a straightforward generalization to uncountable signatures. Vaught, van Heijenoort and Dreben] agree that both the completeness and compactness theorems were implicit in Skolem …. History and Philosophy of Logic. Areas of mathematics. Category theory Information theory Mathematical logic Philosophy of mathematics Set theory.
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