There are exercises in the notes, and I may give others in class.
Corrections to notes and lectures:
- Exercise 3.8 should be to find an n-ary open
formula φ for some n such that the sentence
is a validity, but not a tautology.
- The proof of Theorem 4.1.2 assumes that all structures have
non-empty universes. This is a common assumption in model-theory,
but it is not an assumption that is made explicitly in the notes. If
empty universes are allowed, then the theorem should be understood as
In TO*, every formula φ(x0,…xn−1) is equivalent either to an open formula α(x0,…xn−1) or to ∀x x=x or to ∃x x≠x.
- In §4.2, the theory called Itr* should have
additional axioms, namely, for each positive integer n, an
where x(n) is as defined recursively in the proof of Theorem 4.2.1. (Exercise: where does this proof break down without the additional axioms?)
If T is the theory of an equivalence-relation with two
classes, both infinite (as in §5.4, on p. 66), then
I(T, ℵ0) = 1,
when α > 0. (This corrects a remark in class, December 5.) The main point is that I(T, κ) > 1 when κ is uncountable.
I(T, ℵα) = |α + 1|
- In Exercise 5.8, the theory T should say also that each En-class includes infinitely many En+1-classes.
There are three exams in term: