Metamathematics I Problem Set 7

Due: Friday November 30

Problems marked by ∗ are optional

1) a) (Reduction) Suppose A and B are recursively enumerable. Show that are are recursively enumerable sets A_{0} ⊆ A and B_{0} ⊆ B such that A_{0} ∩ B_{0} = ∅ and A_{0} ∪ B_{0} = A ∪ B.

b) (Separation) Suppose A and B are Π_{1} and A∩B = ∅. Show that there is a recursive C such that A ⊆ C and B ∩ C = ∅. (HINT: Apply a) to ¬A and ¬B.)

2. Show that {e : W_{e} ≠ ∅} is Σ_{1}-complete.

3. Let Cof = {e : ¬W_{e} is finite}.

a) Show that Cof is Σ_{3}.

b)^{∗} Show that Cof is Σ_{3} complete.

4. Suppose X ⊂ N^{2} is Δ_{n}. Show that there is a Δ_{n} set Y ⊆ N such that for all n, Y ≠ X_{n} (where X_{n} = {m : (n, m) ∈ X}). Thus there is no universal Δ_{n} set. (Hint: Consider {n : (n, n) ∉ X}).