Gödel's incompleteness theorems

Agenda

  • Historical context

  • First and second theorems

  • Proof sketch

Logical positivism

Hilbert's program

Principia Mathematica

Along comes Gödel...

The first bombshell

No consistent system of axioms ... is capable of proving all truths about the arithmetic of the natural numbers.

 

For any such formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system.

(Wikipedia)

The second bombshell

The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.

(Wikipedia)

Proof sketch

  • Formal theory
  • Gödel numbering
  • "I cannot be proved"

Formal theory

A formal theory consists of:

  • axioms
  • inference rules

 

It uses a language with a finite set of symbols

 

e.g. Peano arithmetic uses these symbols: 

 

0, S, +, \times, \land, \lor, \lnot, \forall, \exists, =, \lt, (, ), x, *
0,S,+,×,∧,∨,¬,∀,∃,=,<,(,),x,∗0, S, +, \times, \land, \lor, \lnot, \forall, \exists, =, \lt, (, ), x, *

Gödel numbering

Assign a natural number to each symbol in the formal language, e.g.

      = 47

      = 53

      = 84

...

0
00
S
SS
+
++

Gödel numbering

Extend this to sequences of symbols:

     

G(S S S 0) = 2^{53} * 3^{53} * 5^{53} * 7^{47} \approx 10^{118}
G(SSS0)=253∗353∗553∗747≈10118G(S S S 0) = 2^{53} * 3^{53} * 5^{53} * 7^{47} \approx 10^{118}

Thanks to fundamental theorem of arithmetic,

can always go back the other way as well,

from a number to a sequence of symbols

Inference rules can be represented as binary relations on natural numbers

Sequences of symbols,

e.g. formulas and sequences of formulas (proofs),

can be represented by their corresponding Gödel numbers

Results of Gödel numbering

We can come up with the set of the Gödel numbers of all provable statements

The notion of provability in the formal theory can be encoded using Gödel numbers

Through some mind-bending diagonalization tricks,

Gödel comes up with a statement G that says

"the formula whose Gödel number is g is unprovable",

where the formula whose Gödel number is g

turns out to be G itself.

 

In other words, G says "G is unprovable".

"I cannot be proved"

G says "G is not provable".

 

If G is provable, then both a statement and its negation are provable. So the formal system is inconsistent, which violates our hypothesis. Therefore G is not provable.

 

If not(G) is provable, then the formal system turns out not to be ω-consistent, which violates our hypothesis. Therefore not(G) is not provable.

 

G is not decidable,

therefore the formal theory is not complete.

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