Archive for January, 2008

116b- Homework 3

January 29, 2008

Homework 3

Due Tuesday February 5 at the beginning of lecture.

116- Lecture 7

January 29, 2008

We showed that recursive functions and recursive sets are \Sigma_1-representable in {\sf Q}. 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

January 24, 2008

We proved that a set is recursive iff it is \Delta_1-definable.

We showed some elementary properties of the theory {\sf Q}, defined end-extensions, and verified that {\sf Q} is \Sigma_1-complete.

We also defined what it means for a function, or a set, to be represented in a theory.

116b- Homework 2

January 22, 2008

Homework 2

Due Tuesday January 29 at the beginning of lecture.

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

116b- Lecture 5

January 22, 2008

We showed that a function is in {\sf R} iff it has a \Sigma_1-graph. It follows that a set is r.e. iff it is \Sigma_1-definable.

116b- Lecture 4

January 18, 2008

We defined the notion of r.e. sets and proved Gödel’s lemma stating that there is a \Delta_0 coding of finite sequences by numbers. 

116b- Homework 1

January 15, 2008

Homework 1

Due Tuesday January 22 at the beginning of lecture.

116b- Lecture 3

January 15, 2008

We defined Ackermann’s function and showed it is not primitive recursive. We showed that \Delta_0-formulas have a primitive recursive characteristic function, and defined the class {\sf R} of recursive functions.

116b- Lecture 2

January 15, 2008

We defined the class {\sf PR} of Primitive Recursive functions and showed several examples of functions that belong to this class.

116b- Lecture 1

January 15, 2008

We introduced the course, stated the incompleteness theorems, defined {\sf Q} (Robinson Arithmetic), {\sf PA} (Peano or first-order Arithmetic), and {\sf ACA}_0 (the subsystem of second-order Arithmetic given by the arithmetic comprehension axiom).