Math 406

Notes for this course are also available in the photocopy shop in the METU library. The notes were originally based on the lectures from the course of four years ago; but I have made many changes.

There are exercises in the notes, and I may give others in class.

Corrections to notes and lectures:

1. Exercise 3.8 should be to find an n-ary open formula φ for some n such that the sentence

   ∀x₀…∀x_n−1 φ

   is a validity, but not a tautology.
2. 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 follows:

   In TO*, every formula φ(x₀,…x_n−1) is equivalent either to an open formula α(x₀,…x_n−1) or to ∀x x=x or to ∃x x≠x.

3. In §4.2, the theory called Itr* should have additional axioms, namely, for each positive integer n, an axiom

   ∀x x⁽ⁿ⁾≠x

   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?)
4. 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,

   but

   I(T, ℵ_α) = |α + 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.
5. In Exercise 5.8, the theory T should say also that each E_n-class includes infinitely many E_n+1-classes.

There are three exams in term:

1. October 20 (Monday)
   • Solutions
   • Scores
2. November 26 (Wednesday)
   • Solutions
   • Scores
3. December 22 (Monday)
   • Solutions
   • Scores
4. (Final exam) January 20 (Tuesday), 16.30, in M-13 (as scheduled by METU)
   • Solutions
   • Scores

Exams in term are at 17.40. Only your best two scores will count (along with the final-exam score).