1. EITHER: (a) ‘The iterative conception of set states that sets are formed
after their members.’ How are we to understand this temporal language?
What is its significance for the status of the iterative conception?
OR: (b) ‘There is no good reason to accept the axioms of replacement.
They are not motivated by the iterative conception and their benefits can
be had without it.’ Discuss.
2. Outline a theory of transfinite ordinals and illustrate ways in which it differs
from the finite case.
3. Prove Cantor’s theorem. Explain the sense in which your proof is
impredicative.
4. EITHER: (a) What is meant by ‘aleph-null categoricity’? Prove that firstorder complete arithmetic is not aleph-null categorical. How would your
proof break down when applied to second-order Peano arithmetic?
OR: (b) ‘Second-order theories are categorical, and so capture
mathematical practice. Hence second-order logic has a better claim to
be logic than first-order logic.’ Discuss.
5. What is an effectively enumerable set? How can an effectively enumerable
set fail to be decidable? Explain which of the following comprise effectively
enumerable sets:
(i) the tautologies of the propositional calculus
(ii) the sentences of first-order logic with identity that are not logical
truths
(iii) the logical truths of second-order logic
(iv) the theorems of first-order complete arithmetic.
6. Outline a proof of the completeness of first-order predicate calculus
without identity. Explain how to extend your proof to the case with
identity.
7. Show, for a suitable notion of ‘expressive’ (which you should define),
that any axiomatic theory in a sufficiently expressive language is either
unsound or incomplete.
8. Can the notion of computability be given a precise mathematical
definition? If so, how? If not, why not?
9. Was Hilbert’s programme a total failure?
10.In what sense, if any, is ‘countability’ a relative notion?