Stability-TheoryThis 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).
HomeworkIn its own directory.
NotesIn their own directory. (I managed to type what we did so far, as of March 9. Corrections are welcome.
ScheduleMondays, 13.40&ndash 15.30; Wednesdays, 13.40–14.30; İkeda Room.
ContentSome 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.
ReferencesFor background and review: My notes on basic model-theory.
Links in the following list are to reviews:
- Steven Buechler, Essential Stability Theory (Berlin: Springer, Perspectives in Mathematical Logic, 1996)
- David Marker, Model Theory: An Introduction (New York: Springer, Graduate Texts in Mathematics, 2002)
- Anand Pillay, Geometric Stability Theory (Oxford: Clarendon Press, Oxford Logic Guides, 1996)