Computability and Logic

PROBLEMS 285
Problems
22.1 Does it follow from the fact that ∃xFx & ∃x∼Fx is satisfiable that ∃X(∃xXx &
∃x∼Xx) is valid?
22.2 Let us write R ∗ ab to abbreviate
∀X[Xa& ∀x∀y((Xx & Rxy) → Xy) → Xb].
Show that the following are valid:
(a) R ∗ aa
(b) Rab → R ∗ ab
(c) (R ∗ ab & R ∗ bc) → R ∗ ac
Suppose Rab if and only if a is a child of b. Under what conditions do we have
R ∗ ab?
22.3 (A theorem of Frege) Show that (a) and (b) imply (c):
(a) ∀x∀y∀z[(Rxy & Rxz) → y = z]
(b) ∃x(R ∗ xa & R ∗ xb)
(c) (R ∗ ab ∨ a = b ∨ R ∗ ba).
22.4 Write ♦(R) to abbreviate
∀x∀y(∃w(Rwx & Rwy) →∃z(Rxz & Ryz)).
Show that ♦(R) →♦(R*) is valid.
22.5 (The principle of Russell’s paradox) Show that ∃X∼∃y∀x(Xx ↔ Rxy) is
valid.
22.6 (A problem of Henkin) Let Q1 and Q2 be as in section 16.2, and let I be the
induction axiom of Example 22.6. Which of the eight combinations
{(∼)Q1, (∼)Q2, (∼)I }, where on each of the three sentences the negation
sign may be present or absent, are satisfiable?
22.7 Show that the set of (code numbers of) second-order sentences true in the
standard model of arithmetic is not analytical.
22.8 Show that P II is not logically equivalent to any first-order sentence.
22.9 Show that for any first- or second-order sentence A of the language of arithmetic,
either P II & A is equivalent to P II ,orP II & A is equivalent to 0 ≠ 0.
22.10 Show that the set of (code numbers of) second-order sentences that are equivalent
to first-order sentences is not analytical.
22.11 Prove the Craig interpolation theorem for second-order logic.