Due Tuesday February 5 at the beginning of lecture.
We showed that recursive functions and recursive sets are -representable in . We also defined Gödel numberings and exhibited an example of one. This allowed us to define when a theory is recursive. Finally, we proved Gödel’s diagonal lemma.
We proved that a set is recursive iff it is -definable.
We showed some elementary properties of the theory , defined end-extensions, and verified that is -complete.
We also defined what it means for a function, or a set, to be represented in a theory.
Due Tuesday January 29 at the beginning of lecture.
Update: There is a typo in problem 2, it must be .
We showed that a function is in iff it has a -graph. It follows that a set is r.e. iff it is -definable.
We defined the notion of r.e. sets and proved Gödel’s lemma stating that there is a coding of finite sequences by numbers.