Morley's Theorem (model theory)

The set of Morley is a sentence from the model theory , a mathematical branch of logic . It states that a countable theory which, apart from isomorphism, has only one model in an uncountable cardinal number , then also only has one model in every uncountable cardinal number. The theorem was proven by Michael D. Morley in 1962 in his dissertation Categoricity in Power . The theorem and its proof had a lasting effect on model theory. In 2003 Morley received the Leroy P. Steele Prize for this . In 1974 Saharon Shelah generalized the theorem to uncountable theories.

history

A theory is called categorically in a cardinal number if there is only one model of the theory with power apart from isomorphism . A theory is called categorical if it is categorical in any cardinal number. ${\ displaystyle \ kappa}$ ${\ displaystyle \ kappa}$

Jerzy Łoś reported in 1954 that he could only find three types of categorical theories:

Theories that are categorical in any cardinal number,
Theories that are countably categorical but not categorical in any uncountable cardinal number, and finally
Theories that are not countably categorical, but are categorical in any uncountable cardinal number.

Łoś suspected that there were no other options. This conjecture was proven by Morley and generalized by Shelah. This was the beginning of further studies in the field of categoricity and stable theories .

Morley's theorem

Let be a countable theory, that is, a theory about a countable language . Be categorical in an uncountable cardinal number. Then in every uncountable cardinal number is categorical. ${\ displaystyle \ mathrm {T}}$ ${\ displaystyle \ mathrm {T}}$ ${\ displaystyle \ mathrm {T}}$

The proof shows more, namely the following corollary:

Let be a complete theory about a countable language and be . If every model of the thickness - saturated , then each model is saturated. ${\ displaystyle \ mathrm {T}}$ ${\ displaystyle \ alpha> \ omega}$ ${\ displaystyle \ mathrm {T}}$ ${\ displaystyle \ alpha \ \ omega _ {1}}$ ${\ displaystyle \ mathrm {T}}$

(Note: Saturated models of the same thickness are always isomorphic; the first uncountable cardinal number is often referred to in model theory with instead of with .) ${\ displaystyle \ omega _ {1}}$ ${\ displaystyle \ aleph _ {1}}$

proof

The original proof was simplified by Baldwin and Lachlan. At the time of Morley's proof, two key elemmas were not yet known. Today's evidence is divided into two parts, an “up” and a “down” part.

Upwards

Is categorically in , it can be shown that is stable and therefore has a saturated model of power . So all models of power are saturated. ${\ displaystyle T}$ ${\ displaystyle \ omega _ {1}}$ ${\ displaystyle T}$ ${\ displaystyle \ omega _ {1}}$ ${\ displaystyle \ omega _ {1}}$

One lemma says that from an uncountable, unsaturated model, one can build an unsaturated model of power . The proof of this lemma is carried out with a version of the Löwenheim-Skolem theorem generalized to two cardinals , which was not known at the time of Morley's proof. ${\ displaystyle \ omega _ {1}}$

Therefore any uncountable model must also be saturated. Saturated models of the same thickness are isomorphic, therefore is also categorical in every uncountable cardinal number. ${\ displaystyle T}$

Down

Now be categorically in and a model of power . There are -saturated models of power , so -saturated. ${\ displaystyle T}$ ${\ displaystyle \ alpha> \ omega _ {1}}$ ${\ displaystyle {\ mathfrak {B}}}$ ${\ displaystyle \ alpha}$ ${\ displaystyle \ omega _ {1}}$ ${\ displaystyle \ alpha}$ ${\ displaystyle {\ mathfrak {B}}}$ ${\ displaystyle \ omega _ {1}}$

The -stability can again be deduced from the -categoricity. With this, a model of mightiness can be constructed from a mighty model , that no more types over countable sets than realized. However, this must be isomorphic according to the prerequisite , which is -saturated. So it is also -saturated, that is, saturated. Since saturated models of equal thickness are isomorphic, all -mighty models are isomorphic. ${\ displaystyle \ alpha}$ ${\ displaystyle \ omega}$ ${\ displaystyle \ omega _ {1}}$ ${\ displaystyle {\ mathfrak {A}}}$ ${\ displaystyle \ alpha}$ ${\ displaystyle {\ mathfrak {A}}}$ ${\ displaystyle {\ mathfrak {B}}}$ ${\ displaystyle \ omega _ {1}}$ ${\ displaystyle {\ mathfrak {A}}}$ ${\ displaystyle \ omega _ {1}}$ ${\ displaystyle \ omega _ {1}}$

Examples

There are little known natural examples of categorical theories. These are among others:

Totally categorical theories

The theory of empty language has exactly one model in every cardinal number except for isomorphism.

The theory of infinite Abelian groups, in which all elements are of order p (p is a prime), is categorical in any cardinal number.

Countable categorical and uncountable categorical theories

The theory of dense linear order without end points is the theory of as an order structure. Georg Cantor showed that two countable models are isomorphic. However, this theory is not categorical in uncountable cardinal numbers. There are models with points that only have a countable number of points between them and models that always have uncountable number of points between each two points. ${\ displaystyle \ mathbb {Q}}$

Another simpler example is the theory with two equivalence classes , which contains the axioms that there are infinitely many elements in both equivalence classes. A countable model must have a countable number of elements in both classes, an uncountable model only needs to have an uncountable number of elements in one class and can have a countable number or uncountable number of elements in the other equivalence class.

Uncountable categorical and uncountable categorical theories

A classic result by Ernst Steinitz says that two algebraically closed bodies with the same characteristic and the same uncountable thickness are isomorphic, while two countable closed bodies with the same characteristic can differ in their degree of transcendence, which is finite or countable.

A divisible torsion-free Abelian group can be understood as a vector space. There are many countable models (depending on the dimension) and exactly one model in uncountable cardinal numbers. ${\ displaystyle \ mathbb {Q}}$

The complete theory of where is the successor function. ${\ displaystyle <\ omega, S>}$ ${\ displaystyle S}$

The theory about language with the axioms that say that it is interpreted differently than . ${\ displaystyle \ {c_ {i} | i <\ omega \}}$ ${\ displaystyle i <j}$ ${\ displaystyle c_ {i}}$ ${\ displaystyle c_ {j}}$

generalization

Shelah managed to generalize Morley's results to uncountable languages: (In doing so, he expanded on earlier work by Rowbottom and Ressayre.)

If a complete theory about a language is categorical in a cardinal number , it is categorical in every cardinal number greater than . ${\ displaystyle L}$ ${\ displaystyle \ alpha> \ | L |}$ ${\ displaystyle | L |}$

References and comments

↑ On the Steele Prize for Morley, Notices of the AMS, 2003, (PDF file) (269 kB)
↑ This definition of categorical follows: Philipp Rothmaler: Introduction to the model theory , Spektrum Akademischer Verlag 1995, ISBN 978-3-86025-461-5 , section 8.5. The original definition of categorical required for categoricity that there is only one model apart from isomorphism, which according to the Löwenheim-Skolem theorem only applies to complete theories with (exactly) one finite model. (O. Veblen, 1904, see Wilfrid Hodges: First Order Model Theory. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy . )
↑ Jerzy Łoś: On the categoricity in power of elementary deductive systems and related problems , Colloq Math., 3 (1954), pp. 58-62.
↑ http://plato.stanford.edu/archives/sum2005/entries/modeltheory-fo/#Thms
↑ Chang, Chen C., Keisler, H. Jerome: Model theory. Third edition. Studies in Logic and the Foundations of Mathematics, 73. North-Holland Publishing Co., Amsterdam, 1990. ISBN 0-444-88054-2 , section 7.1

Web links

Martin Ziegler: Script Model Theory 1 . (PDF; 649 kB)

literature

Michael Morley: Categoricity in power , Transactions of the AMS, Vol. 114, 1965, pp. 514-538
Chang, Chen C., Keisler, H. Jerome: Model theory. Third edition. Studies in Logic and the Foundations of Mathematics, 73. North-Holland Publishing Co., Amsterdam, 1990. ISBN 0-444-88054-2

[1] On the Steele Prize for Morley, Notices of the AMS, 2003, (PDF file) (269 kB)

[2] This definition of categorical follows: Philipp Rothmaler: Introduction to the model theory , Spektrum Akademischer Verlag 1995, ISBN 978-3-86025-461-5 , section 8.5. The original definition of categorical required for categoricity that there is only one model apart from isomorphism, which according to the Löwenheim-Skolem theorem only applies to complete theories with (exactly) one finite model. (O. Veblen, 1904, see Wilfrid Hodges: First Order Model Theory. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy . )

[3] Jerzy Łoś: On the categoricity in power of elementary deductive systems and related problems , Colloq Math., 3 (1954), pp. 58-62.

[4] ttp://plato.stanford.edu/archives/sum2005/entries/modeltheory-fo/#Thms

[5] Chang, Chen C., Keisler, H. Jerome: Model theory. Third edition. Studies in Logic and the Foundations of Mathematics, 73. North-Holland Publishing Co., Amsterdam, 1990. ISBN 0-444-88054-2 , section 7.1