[an integer] [a program] [a code of x] [x halts on y]
[x succeeds/succeed on y and z] [x decides halting]

Axiom. Every program has a code.

Axiom. Every code of every program is an integer.

Definition. Let W be a program.
W decides halting iff
for every code U of every program Y and every integer Z
W succeeds on U and Z iff Y halts on Z.

Axiom. Let W be a program that decides halting.
Then there exists a program V such that for every integer Y
V halts on Y iff W does not succeed on Y and Y.

Proposition. No program decides halting.
