Lecturer: Dr. C. Ó Dúnlaing
Date: 1996-97
Groups: JS and SS Mathematics
Prerequisites: None
Duration: 18 weeks
Lectures per week: 3
Assessment: homework, final exam
Examinations: One 3-hour examination - end of year
Review of some computability theory. Propositional logic,
completeness of SC (sentential calculus). First-order logic,
interpretations, completeness of Predicate Calculus.
Löwenheim-Skolem theorems, Ultraproduct constructions of nonstandard
models. Arithmetic theories, primitive recursive functions, and
Gödel numbering. Gödel's first incompleteness theorem and the
Gödel-Rosser theorem. The Hilbert-Bernays derivability conditions
and Gödel's second incompleteness theorem.
Zermelo-Fraenkel set theory (ZF): ordinal numbers, well-founded sets,
relative consistency of foundation axiom. Axiom of choice and
equivalents. Constructible sets and the relative consistency of the
constructability axiom. Deduction of relative consistency of axiom of
choice and general continuum hypothesis. Fraenkel-Mostowski models and
independence of axiom of choice (without foundation axiom).