I know that we can prove a program is uncountable by basing it off the halting program, which does not exist. To clarify, we can use the theorem
The Halting Problem is uncomputable; i.e., there does not exist a computer program TestHalt with the behavior specified above on all inputs (P, x).
But I don't really get how to put it into practice.
How would you say a program P(F, x, y) that returns true if the program F outputs y when given x as input (i.e. F(x) = y) and false otherwise can't exist?
Suppose such program $P$ existed, but change it a little to output "yes" or "no" instead of true and false. Let $G$ be the program that outputs 1 on any input, and for any program $F$, let $F_1 = G\circ F$. Then $F_1$ is computable, so the function $H(x) = P(F_1, x, 1)$ is also computable. But this function is exactly $\text{TestHalt}(F,x)$, a contradiction.