Morley's categoricity theorem
The set of Morley is a set of model theory, a branch of mathematical logic. He says that a countable theory in a uncountable cardinal number has only one model up to isomorphism, then in every uncountable cardinal number has only one model. Has been proved to the set of Michael Morley in 1962 in his dissertation Categoricity in Power. The theorem and its proof exerted a lasting effect on the model theory. In 2003 Morley was awarded the Leroy P. Steele Prize. Saharon Shelah generalized in 1974 set on uncountable theories.
- 4.1 Total categorical theories
- 4.2 denumerable categorical and uncountably categorical theories not
- 4.3 Non- categorical countable and uncountable categorical theories
History
A theory is called categorical in a cardinal number, if there is only one model of the theory with thickness up to isomorphism. A theory is called categorical if it is categorical in some cardinal number.
Jerzy Łoś reported in 1954 that he could find only three types of categorical theories:
Łoś suspected that there are no other possibilities. This conjecture was proved by Morley and generalized by Shelah. This was the beginning of further investigations in the field of categoricity and the stable theories.
Set of Morley
Be a countable theory, ie a theory of a countable language. Let further categorically in an uncountable cardinal number. Then is categorical in every uncountable cardinal number.
The proof shows more, namely the following corollary:
Be a complete theory over a countable language and let. If each model is - saturated thickness of the, then each model is saturated by.
(Note: Saturated models of the same cardinality are always isomorphic, the first uncountable cardinal number is often referred to in the model theory instead. )
Evidence
The original proof could be simplified by Baldwin and Lachlan. At the time of Morley's proof two Schlüssellemmata were not yet known. The present proof is divided into two parts, one "up" - and a " down " part.
Is categorically, so it can be shown that is stable and therefore has a saturated model of cardinality. So all models of cardinality are saturated.
A lemma says that you can not form a saturated model of cardinality of an uncountable not saturiertem model. The proof of this lemma is done with a generalized two cardinal numbers version of the theorem of Löwenheim - Skolem, which was not known at the time of Morley's proof.
Therefore, any überabzählbares model must be saturated. Saturated models of the same cardinality are isomorphic, therefore, is also categorically in every uncountable cardinal number.
Down
Now let categorically and a model of cardinality. There - saturated models of cardinality, ie is - saturated.
From - categoricity on the stability can be closed again. This can be constructed from a powerful model - a model of cardinality, that not more types over countable sets as realized. This must, however, be isomorphic by assumption, which is - saturated. So well - saturated, so saturated. Since saturated models of the same cardinality are isomorphic, all - powerful models are isomorphic.
Examples
There is little known natural examples of categorical theories. These include:
Total categorical theories
- The theory of the empty language has in every cardinal number up to isomorphism exactly one model.
- The theory of infinite abelian groups, in which all elements have order p (p is prime ) is categorical in every cardinal number.
Countably categorical and uncountably categorical theories not
- The theory of dense linear order without endpoints is the theory of the structure of order. Georg Cantor showed that two countable models are isomorphic. However, this theory is not categorical in uncountable cardinals. So there are models with points that have only a countable number of points between them and models that always have uncountably many points between any two points.
- Another simpler example is the theory with two equivalence class that includes the axioms that are an infinite number of elements in both equivalence classes. A countable model must in both classes have a countable number of elements, a überabzählbares model must only be uncountable in a class have many items and can in the other equivalence class is countable or uncountable many have many items.
Not countably categorical and uncountably categorical theories
- A classical result of Ernst Steinitz says that two algebraically closed fields of the same characteristics and the same uncountable cardinality are isomorphic, while two countable closed fields of the same characteristics can differ by their transcendence degree which is finite or countable.
- A divisible torsion-free abelian group can be seen as vector space. There are countably many countable models (depending on dimension) and in uncountable cardinal numbers exactly one model.
- The complete theory of, the successor function.
- The theory of language with the axioms, which state that for other than is interpreted.
Generalization
Shelah was able to generalize Morley's results on uncountable languages: ( He extended this earlier work by Rowbottom and Ressayre. )
Is a complete theory over a language categorically in a cardinal number, it is categorical in every cardinal number that is greater than.