Gödelization

example

third Peano axiom part a

expression:

(Sa=Sa')(a=a')

Gödel number:

1

2

×

15

3

×

11

5

×

17

7

×

15

11

×

11

13

×

19

17

×

9

19

×

13

23

×

1

29

×

11

31

×

17

37

×

11

41

×

19

43

×

9

47

=2×14348907×48828125×232630513987207×4177248169415651×1792160394037×239072435685151324847153×322687697779×504036361936467383×29×25408476896404831×456487940826035155404146917×550329031716248441×10861771343660416614908294685907×1119130473102767

In 1931 Kurt Gödel established a representation between a formal system and the set of natural numbers to prove his famous incompleteness theorem. An axiom or a proof is encoded by assigning to each symbol in the expression odd numbers as the powers of successive primes. The expression can be recovered by factoring the number.