PROOFS

Claim: There exists an irrational number X and an irrational number Y such that X to the power of Y is rational.

Proof:

Let X = sqrt(2)
Let Z = sqrt(2) to the power of sqrt(2)

Case 1: Z is rational (we have proved that there exists a irrational number X and irrational number Y such that X to the power of Y is rational. X = sqrt(2) and Y = sqrt(2). )

Case 2: Z is irrational. Then Z to the power X = 2 which is rational -- proved

We do not know which one is true case or case 2; we don't care. We have a proof.

Claim: There exists a program which will tell if the President will be sitting in a chair or not sitting in a chair tomorrow 10 PM?

Proof:

Two programs:
Program 1: Outputs He is sitting.
Program 2: Outputs He is not sitting.

One of Program 1 or Program 2 is the right one. We don't know which one. But does not matter.

There exists a program ...

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Claim: There exist a function that takes natural numbers (1,2, 3 ...) as input and outputs Yes or No which cannot be computed by a Java/C/C++/etc. program.

Proof: Let uss consider all Java programs that take natural numbers (1,2, 3 ...) as input and outputs Yes or No.

Lets sort these Java program alphabetically and call the first one J1, the second one J2 and so on.
Now consider the following function F:

Input(X: Natural Number)

If J_X on input X says YES then Output NO

elseif J_X on input X says NO then Output YES

Claim: The above function cannot be computed by a Java program.

It can not be computed by J1 as F(1) != J1(1)

It can not be computed by J2 as F(2) != J2(2)

It can not be computed by J230 as F(230) != J230(230)