Math 564: Model Theory II

I am Stephen G. Simpson, a Professor of Mathematics at Penn State University.

Math 564 is a graduate course on model theory. I am teaching it in Fall 2004.

The textbook is Model Theory: An Introduction, by David Marker, Springer-Verlag, Graduate Texts in Mathematics, 2002.

A course announcement is here in several formats: PS, PDF, DVI.

Also of interest is the Penn State Logic Seminar.

Homework:

Chapter 1, Exercises 1.4.5, 1.4.8, 1.4.14, 1.4.15, 1.4.16.
Note: In Exercise 1.4.5, a tree is defined to be a partial ordering with a least element, in which the predecessors of each element are linearly ordered.
Chapter 2, Exercises 2.5.3, 2.5.4, 2.5.5, 2.5.11, 2.5.27, 2.5.28, 2.5.33. Please note the following corrections to the problem statements.
- In Exercise 2.5.3, instead of "language with one binary relation symbol", say "language containing a binary relation symbol".
- In Exercise 2.5.4, instead of "infinitely many infinite classes", say "infinitely many classes, all of which are infinite".
- In Exercise 2.5.11, prove in addition that you can get the intersection of the images of M_1 and M_2 in N to be just the image of M_0 in N.
- In Exercise 2.5.28, instead of "dense linear orders", say "dense linear orders without end points".
- In Exercise 2.5.28, in part (c), add the sentences P(c_n), n = 0, 1, 2, ....
Chapter 4, Exercises 4.5.1, 4.5.2, 4.5.4, 4.5.5, 4.5.7, 4.5.11, 4.5.16, 4.5.22, 4.5.31, 4.5.45, plus the following variants of Exercises 4.5.7, 4.5.15, 4.5.35, 4.5.39 respectively.
- Let T be a complete extension of ZFC. Show that | S_1(T) | is of cardinality 2^aleph_0.
- Construct a model M of ZFC of cardinality aleph_1 such that for all a in M, {b in M | b in_M a} is countable.
- Let M be a saturated L-structure, where L is countable and M is of cardinality 2^kappa = kappa^+. Show that for any countable language L* extending L and any L*-theory T* consistent with Th(M), there exists a saturated model M* of T* such that M = M* | L, the L-reduct of M*.
- Assuming the GCH, use saturated models to prove Robinson's Consistency Theorem: Let L = L_1 intersect L_2, and let T be a complete L-theory included in T_1 intersect T_2 where T_1 is an L_1-theory and T_2 is an L_2 theory, then T_1 union T_2 is consistent.

t20@psu.edu   /   15 November 2004