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Stability-Theory

This is a course in logic with connexions to algebra, topology and geometry. It follows Math 736, whose web-page see for background reading (lecture-notes for Math 406 and 736; notes on ultraproducts; notes on Galois correspondences).

Homework

In its own directory.

Notes

In their own directory. (I managed to type what we did so far, as of March 9. Corrections are welcome.

Schedule

Mondays, 13.40&ndash 15.30; Wednesdays, 13.40–14.30; İkeda Room.

Content

Some rough and incomplete remarks:

Stability-theory is the part of model-theory based on the following definition: A complete (first-order) theory with infinite models is called κ-stable (for some infinite cardinal κ) if it has no more than κ-many complete types over any parameter-set of size κ.

The first big theorem of the subject is Morley's Categoricity Theorem: If a countable theory is categorical in some uncountable cardinality, then in all. The proof involves showing that the hypothesis and conclusion are equivalent to being ω-stable and having no “Vaughtian pairs”.

As a result of Morley's Theorem, we can refer unambiguously to uncountably categorical countable theories. Natural examples include the theory of algebraically closed fields of a given characteristic and the theory of vector-spaces over a given countable field.

A general motivating principle is that categorical theories are expected to be mathematical interesting or tractable theories.

References

For background and review: My notes on basic model-theory.

Links in the following list are to reviews:

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Last change: 2005, March 18