Table of contents 
2 Step 2 3 Step 3 4 An informal account of the proof 
Gödel numbers are constructed with reference to symbols of the propositional calculus and formal arithmetic. Each symbol is first assigned a natural number, thus:
Logical symbols  Numbers 1:12  

¬ ( ) S 0 = . +

1 ("not") 2 ("for all") 3 ("if,then") 4 ("or") 5 ("and") 6 7 8 ("is the successor of") 9 10 11 12  
Propositional symbols  Numbers greater than 10 but divisible by 3  
P Q R S 
12 15 18 21 

Individual variables  numbers greater than 10 with remainder 1 when divided by 3  
v x y 
13 16 19 

Predicate symbols  numbers greater than 10 with remainder 2 when divided by 3  
E F G 
14 17 20 
Arithmetical statements are assigned unique Gödel numbers referenced to the series of prime numbers. This is based on a simple code which essentially reads
prime 1 ^{character 1} * prime 2 ^{character 2} * prime 3 ^{character 3}
and so on.
For example the statement
x, P(x) becomes
2^{2} * 3^{16} * 5^{12} * 7^{6} * 11^{16} * 13^{7}, because
{2, 3, 5, 7, 11, ...} is the prime series, and 2, 16, 12, 6, 16, 7 are the relevant character codes. This is a large but perfectly determinate number (on the order of 10^46).
Finally, sequences of arithmetical statements are assigned further Gödel numbers, such that the sequence
Statement 1 (GN1)
Statement 2 (GN2)
Statement 3 (GN3)
gets the Gödel number 2^{GN1}*3^{GN2}*5^{GN3}, which we will call GN4. The proof of Gödel's incompleteness theorem depends on the demonstration that, in formal arithmetic, some sets of statements logically entail or prove other statements. For example it might be shown that GN1, GN2, and GN3 together, i.e. GN4, prove GN5. Because this is a demonstrable relationship between numbers it is entitled to its own symbol, for example R. R(v,x) would then mean "x proves v". In the case where x and v are Gödel numbers GN5 and GN4 we would say
R(GN5,GN4).