Lecture 17

Today:

    1. Quiz.

    2. Godel numbering.

    3. Godel's proof.

For next Monday:

    1. Read Nagel and Newman pages 68-80.

    2. Finish Homework 9.

    3. Prepare for an exam.

For next Thursday:

    1. Finish reading Nagel and Newman.

    2. Prepare the reading questions below.

What's on the exam?

Probably:

    1. Prolog.

    2. Sipser Chapters 2, 3, and 4.

    3. Casti.

Less likely, but not strictly out of bounds:

    1. Racket.

    2. Sipser, Chapter 1.

Godel's Proof

First incompleteness theorem: in any consistent FAS where all derivable strings map to true statements about arithmetic, there are true statements about arithmetic that cannot be derived in the FAS.

Second incompleteness theorem: in any FAS that can be mapped onto statements about arithmetic,

a) if it is possible to derive the statement that the FAS is consistent, then the FAS is inconsistent.

b) if the FAS is consistent, then it is not possible to derive a statement of its consistency.

Big Idea #1: arithmetize syntax

construct a mapping between strings in the FAS and numbers.

Big Idea #2: arithmetize meta-math

construct a string in the FAS that maps to a statement that means "this string can not be derived"

this string is the Godel sentence, denoted G.

Now there are only two possibilities:

If G cannot be derived, then Interp(G) is true, but that means it is a true statement that cannot be derived;

hence the FAS is not complete.

If G can be derived, then ~G can also be derived, which means that the FAS is inconsistent.

If we assume that arithmetic is consistent, or if we prove by other means that it is, then neither G nor ~G can be derived, and so G is "formally undecidable".

This implies that if arithmetic is consistent, then Interp(G) is true.

That was the first incompleteness theorem.

So all this is left is

1) how to construct G.

2) how to get from the first theorem to the second.

That's where Nagel and Newman come in.

.

Reading questions

Nagel and Newman and Hofstadter, Godel's Proof, pages 68-108.

1) What's a Godel number?

2) What are the three kinds of variables that can be encoded?

3) If you had to decode a Godel number, how would you distinguish between a sequence of symbols and a sequence of formulas?

4) Is every number the Godel number of some expression?

5) What arithmetic operation can be used to check whether one formula is a prefix of another?

6) What is the meta-mathematical statement associated with Dem(x, y)?

7) If you evaluated the expression Sub(y, 17, y), what kind of thing would you get (string, number, meta-mathematical statement)?

8) What meta-mathematical process does Sub(y, 17, y) describe?

9) What is the relationship between formula (1) and formula (G)?

10) What is the Godel number of formula (G)?

11) What is the meta-mathematical statement associated with (G)?

12) Why do N&N say that PM is "essentially incomplete"?

13) What is the meta-mathematical interpretation of formula (A)?

From now on, let's choose the branch that says that the meta-mathematical statement "Arithmetic is consistent" is true.

14) How do we know that G is true?

15) How do we know that A->G is true?