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, ....

- In Exercise 2.5.3, instead of "language with one binary
relation symbol", say "language containing a binary relation
symbol".
- 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.

- Let T be a complete extension of ZFC. Show that | S_1(T) | is
of cardinality 2^aleph_0.

t20@psu.edu / 15 November 2004