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
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