116b- Homework 3

Homework 3

Due Tuesday February 5 at the beginning of lecture.

116- Lecture 7

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.

116b- Lecture 6

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.

116b- Homework 2

Homework 2

Due Tuesday January 29 at the beginning of lecture.

 Update:  There is a typo in problem 2, it must be .

116b- Lecture 5

We showed that a function is in iff it has a -graph. It follows that a set is r.e. iff it is -definable.

116b- Lecture 4

We defined the notion of r.e. sets and proved Gödel’s lemma stating that there is a coding of finite sequences by numbers. 

116b- Homework 1

Homework 1

Due Tuesday January 22 at the beginning of lecture.

116b- Lecture 3

We defined Ackermann’s function and showed it is not primitive recursive. We showed that -formulas have a primitive recursive characteristic function, and defined the class of recursive functions.

116b- Lecture 2

We defined the class of Primitive Recursive functions and showed several examples of functions that belong to this class.

116b- Lecture 1

We introduced the course, stated the incompleteness theorems, defined (Robinson Arithmetic), (Peano or first-order Arithmetic), and (the subsystem of second-order Arithmetic given by the arithmetic comprehension axiom).