1. ‘It is impermissible to use the language of second-order logic to
formalise discourse about certain sorts of objects, such as sets or
ordinals, in case there is no set to which all objects of that sort belong.’
Discuss.
2. Explain the role of Henkin constants (also known as witnesses) in the
proof of the completeness of first-order predicate logic without identity.
What further complexities are introduced in proving completeness for
the system with identity? What role does mathematical induction play
in these proofs?
3. What is meant by a ‘non-standard model’? Do such models have any
philosophical significance?
4. How do cardinals and ordinals differ?
5. What is the iterative conception of set? Which axioms of ZFC does it
justify?
6. In what sense, if any, does set theory provide a foundation for ordinary
mathematics?
7. ‘An algorithm is a procedure for which we can give exact instructions
for carrying it out.’ Discuss.
8. “No formal system for arithmetic can prove its own consistency.” Is this
correct? If not, why not?
9. ‘PA can prove its own consistency. This is because PA can capture
ZFC as a deductive system and ZFC can prove the consistency of
PA.’ What is wrong with this reasoning? How does your answer bear
on the question of whether minds are machines?
10. How far can Hilbert’s programme be taken?