Godel’s proof

Danny Brown

Outline of godel’s proof

1. Create a statement that says of itself that it is not provable

2. Show that this statement is provable iff it is not provable (Richard’s paradox)

3. Conclude that if our system is consistent then this statement is undecidable

4. Surprise

Godel numbering

• Every symbol has a number • Every statement has a number• Every series of statements has a number

• So statements about arithmetic can be described by arithmetical relationships

Godel numbering (Nagel and Newman)

Suppose ‘0’ is given the value (godel number) 21, ‘=‘ has godel number 36, then:

Statement ‘0=0’ has godel number

213621 532

Uniqueness

• So every statement has a unique Godel number

• Given a number you could factorise to find the statement

The proof

proof[a,b]

This formula means a proves b

a is actually the number representing some (sequence of) statements that proves the statement with number b

It’s really states that there is an arithmetical relation between the numbers for a and b.

sub(n,21,c)

This formula means we substitute ‘c’ for the symbol with value 21 in the statement with number n

eg in the statement ‘0=0’ (with number n) we could substitute the symbol ‘0’ (=21) with the symbol ‘c’…

then the statement becomes ‘c=c’…

…if c=71 (say) then the new statement has value

sub(n,21,c) = 713671 532

Creating the statement G

Consider the statement

(this means that b is not provable)

baproofa ,)(

Creating the statement G

Now consider the following statement

Let this statement have value m

),71,(,)( ccsubaproofa

Creating the statement G

Now substitute the symbol m in place of c in the statement to give

Let’s call the value of this statement g… but what does g equal??

),71,(,)( mmsubaproofa

g = ?

substitute the symbol m for the symbol with value 71 (=c) into the statement with value m…

so g = ?

g = ?

substitute the symbol m for the symbol with value 71 (=c) into the statement with value m…

so g = sub(m,71,m)

What does G say?

Look closely at the statement G again…

),71,(,)( mmsubaproofa

What does G say?

Look closely at the statement G again…

…or…

…what does it say??

),71,(,)( mmsubaproofa

gaproofa ,)(

Is G provable or not?

Is G provable or not?

G says of itself that it is not provable

…so…

Is G provable or not?

G says of itself that it is not provable

…so… G is provable iff it is not provable

Consistency

We can’t have statements in our logical system that are both provable and not provable (not ‘consistent’)

So we keep consistency and conclude that this sentence is neither ie it is ‘undecidable’ (and the system is ‘incomplete’).

Surprise

Does it matter that G is undecidable?

Is G true?

What does Godel’s theorem say?

There are always undecidable arithmetical statements in any ‘sufficiently expressive’ arithmetical system of logic.

Self-reference

Objection: only self-referential statements are the ones that can’t be proved…

‘An introduction to Godel’s Theorems’ Peter Smith

What it doesn’t say

• There are ‘unprovable truths’

- it is independent of what we think is true or not (although we can introduce the idea of soundness)

- given different sets of axioms, some statements will be provable and some won’t… maybe mathematical axioms could change in the future?

But which theorems are they?

• The Collatz conjecture?• Goldbach’s?• Riemann Hypothesis?

…cue Alan Turing