Peter Koepke: Computations with Ordinals, and Models of Set Theory
Ordinary computations can be characterised by register machines working
with natural numbers. We introduce ordinal register machines where the
registers can hold arbitrary ordinal numbers. The class of sets of
ordinals which are computable by such machines has strong closure
properties. It allows to formalise the usual mathematical notions like
relations and functions, numbers, and infinitary recursion. It can thus be
used as a foundation of mathematics and it codes Gödel's model of
constructible sets.
A tentative course outline is as follows:
- Review of the theory of ordinals.
- Review of standard register machines.
- Definition of ordinal register machines.
- Notions of ordinal computability.
- A universal ordinal machine.
- Closure properties of the class of ordinal computable sets.
- The theory of sets of ordinals.
- Relations to Gödel's constructible model L
A first introduction to ordinal computations (with Turing rather than
register machines) can be found in
P. Koepke, Turing computations on ordinals, The Bulletin of Symbolic Logic
11, No. 3, pp. 377-397.
Further material will be made available during the course.
Reinhard Kahle, 21.01.06