Due Tuesday February 5 at the beginning of lecture.

## Archive for January, 2008

### 116b- Homework 3

January 29, 2008### 116- Lecture 7

January 29, 2008We 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

January 24, 2008We 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

January 22, 2008Due Tuesday January 29 at the beginning of lecture.

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

### 116b- Lecture 5

January 22, 2008We 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

January 18, 2008We 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

January 15, 2008Due Tuesday January 22 at the beginning of lecture.

### 116b- Lecture 3

January 15, 2008We 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

January 15, 2008We defined the class of Primitive Recursive functions and showed several examples of functions that belong to this class.

### 116b- Lecture 1

January 15, 2008We 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).