Solutions of exercises in Chapter 7
E7.1 Let M be a eld, A a subeld, and a M . Suppose that a is algebraic over A in the
usual sense of eld theory. Show that a is algebraic over A in the model-theoretic sense.
Let f (x) be a polynomial with coecients in A
Solutions of exercises in Chapter 6
E6.1 A subset X of a structure M is denable iff there is a formula (x) with only x
free such that X = cfw_a M : M |= [a]. Similarly, for any positive integer m, a subset
X of m M is denable iff there is a formula (x) wi
Solutions of exercises in Chapter 5
E5.1 Suppose that A is an L -structure. Let F be a nonprincipal ultralter on a set I .
For each a A let f (a) = [ a : i I ]F . Show that f is an embedding of A into I A/F ,
and A is elementarily equivalent to I A/F .
Fo
Solutions of exercises in Chapter 4
4.1 Prove Proposition 4.7.
For (i), we proceed by induction on . If is vj with j = i, or is an individual constant,
then = and so is a term. Suppose that is vi . Then = if 0 occurrences of vi are
replaced, or is is vi i
Solutions of exercises in Chapter 3
3.1 Dene | = . (The Sheer stroke.). Show that and can be dened in
terms of |.
is equivalent to |. Then is equivalent to ()|( ).
3.2 A formula involving only S0 , . . . , Sm determines a function t : m+1 2 2 dened
m
by
Solutions of exercises in Chapter 2
2.1 Prove that for any class K of algebras we have SHK HSK.
Suppose that M SHK. Then there are N K and P such that there is a homomorphism
f from N onto P and M is a subalgebra of P . Let Q = f 1 [M ]. Clearly Q is a su
Solutions of exercises in Chapter 1
1.1 Let L be a language with no individual constants. Dene an L -structure A and
subuniverses B, C of A such that B C = .
Let A = 2, and let the fundamental operations of A be such that cfw_0 and cfw_1 are closed
under
7. Morleys theorem
This chapter is devoted to the proof of Morleys theorem, which says that in a countable
language, if is a theory with only innite models and is -categorical for some uncountable cardinal , then it is -categorical for every uncountable c
6. Basic model theory
We survey important notions and results in model theory.
Isomorphisms
Theorem 6.1. Suppose that h is an isomorphism from A onto B , where these are
L -structures. Suppose that a A, is a term, and is any formula. Then h( A (a) =
B (h
5. The compactness theorem
Here we prove the compactness theorem: If a set of sentences is such that every nite
subset of it has a model, then the whole set has a model. The theorem will be an easy
consequence of Los theorem on ultraproducts.
s
Theorem 5.
4. First-order logic
In this chapter we nish introducing the notion of rst-order logic, and connect this notion
to satisfaction and truth in structures.
Let a signature = (Fcn, Rel, Cn, ar) be given. Now in addition to the variables and
the logical symbol
3. Sentential logic
Here we discuss some more components of our nal rst-order logic: the logic surrounding
words like not, and, etc. The language here is simpler than what we have dealt with
so far. We have only the following symbols:
n, a symbol for nega
2. Terms and varieties
We now make one important step towards full model theory: terms and equations. Let be
any signature. We assume given now a simple innite sequence v0 , v1 , . . . of distinct objects
called variables, dierent from the relation symbol
Model theory
(Math 6000)
November 19, 2012
These notes form an introduction to model theory. The topics are: rst-order structures;
terms and varieties; rst-order languages, satisfaction, and truth; elimination of quantiers; Lwenheim-Skolem theorems; ultra