Gödel's Incompleteness: Explorer

Step 1 · The alphabet

G ⇔ ¬Prov(⌜G⌝)

The alphabet of arithmetic

Give every symbol a number.

Pick any reasonable convention. Here are eleven symbols of a simple arithmetic language, each assigned a natural number. Once symbols are numbers, sequences of symbols (i.e. formulae) are sequences of numbers — and any sequence of numbers can be packed into a single natural number.

not

code 1

∨

code 2

∀

for all

code 3

∃

there exists

code 4

equals

code 5

zero

code 6

successor

code 7

plus

code 8

(

open

code 9

)

code 10

variable

code 11

Next: pick a small formula like 0 = 0 or ¬(0 = 0) from the sidebar and watch its Gödel number get built.