Gödels Incompleteness Theorems Kurt Gödel His Theorems A ...

Introduction Gödel Numbering Meta-mathematics Gödels Proofs Conclusions
Map variables to numbers
Introduction Gödel Numbering Meta-mathematics Gödels Proofs Conclusions
Map formulae to numbers
There are three kinds of variables:
◮ Numerical variables (x, y, z): ranging over numerals
0, s0, ... or numerical expressions like x + y
◮ Sentential variables (p, q, r): ranging over closed formulae
(sentences)
◮ Predicate variables (P, Q, R): ranging over predicates
Type Examples Gödel numbers
Numerical x, y, z, . . . 13, 17, 19, . . .
Sentential p, q, r, . . . 13 2 , 17 2 , 19 2 , . . .
Predicate P, Q, R, . . . 13 3 , 17 3 , 19 3 , . . .
( ∃ x ) ( x = s y )
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
8 4 13 9 8 13 5 7 17 9
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
m = 2 8 ×3 4 ×5 13 ×7 9 ×11 8 ×13 13 ×17 5 ×19 7 ×23 17 ×29 9
We call m the Gödel number of (∃x)(x = sy).
Gödels Incompleteness Theorems (Sebastian Bader ) 9
Introduction Gödel Numbering Meta-mathematics Gödels Proofs Conclusions
From numbers to formulae
Gödels Incompleteness Theorems (Sebastian Bader ) 10
Introduction Gödel Numbering Meta-mathematics Gödels Proofs Conclusions
Map proofs to numbers
Can we go backwards? Yes if a number is a valid Gödel
number then we can construct the corresponding formula:
243,000,000
↓
2 6 3 5 5 6
↓ ↓ ↓
6 5 6
↓ ↓ ↓
0 = 0
Formula
(∃x)(x = sy)
(∃x)(x = s0)
Gödel number
m
n
The Gödel number of the (part of the) proof
is k = 2 m × 3 n .
(∃x)(x = sy)
(∃x)(x = s0)
Gödels Incompleteness Theorems (Sebastian Bader ) 11
Introduction Gödel Numbering Meta-mathematics Gödels Proofs Conclusions
Properties
Gödels Incompleteness Theorems (Sebastian Bader ) 12
Introduction Gödel Numbering Meta-mathematics Gödels Proofs Conclusions
The Correspondence Lemma
◮ For every formula in PM there is a unique Gödel number.
◮ For every proof in PM there is a unique Gödel number.
◮ The Gödel-number-function and its inverse are
computable.
Lemma
Every primitive recursive truth, when represented as a string of
symbols, is a theorem of PM.
Gödels Incompleteness Theorems (Sebastian Bader ) 13
Introduction Gödel Numbering Meta-mathematics Gödels Proofs Conclusions
A simple typographic property
Gödels Incompleteness Theorems (Sebastian Bader ) 14
Introduction Gödel Numbering Meta-mathematics Gödels Proofs Conclusions
A simple typographic property, ctd
◮ Gödel numbers map formulae / proofs to natural numbers.
◮ But we can also express statements about structural
properties using numbers.
Example
The first symbol of the formula “¬(0 = 0)” is “¬”.
Example
The first symbol of the formula “¬(0 = 0)” is “¬”.
◮ How can this be expressed using arithmetics?
◮ The Gödel number of the formula ¬(0 = 0) is
a = 2 1 × 3 8 × 5 6 × 7 5 × 11 6 × 13 9 .
◮ We can state that the exponent of 2 (first position, hence
smallest prime) in a’s prime factorisation is 1 (Gödel
number of ¬ = 1).
◮ In other words: 2 is a factor of a, but 2 2 is not.
Gödels Incompleteness Theorems (Sebastian Bader ) 15
Gödels Incompleteness Theorems (Sebastian Bader ) 16