Does there exist a universal procedure by which we can determine all mathematical truths?

THEOREM:

No such universal procedure can exist.

 

PROOF BY CONTRADICTION:

Suppose there does exist a Universal Truth Machine (UTM), that can carry out the universal procedure to determine the truth of all knowable truths. It will say whether a given statement is true or false.

When I come to this machine I will ask for its program and circuit design. This program may be complicated, but it will be only finitely long. Call this program P(UTM).

I smile, writing for it the following sentence:

"The machine constructed on the basis of the program P(UTM) will never say this sentence is true."

Call this sentence G, in honor of Godel. This is equivalent to:

G = UTM will never say G is true.

Now, I laugh my high laugh and ask the UTM the truth-value of G: whether or not G is true.

If UTM says G is true, then "UTM will never say G is true" is false. If G is false and UTM says that G is true, UTM has made a false statement. So, UTM will never say that G is true, since UTM makes only true statements.

It is established that UTM will never say G is true. So "UTM will never say G is true" (G) is in fact a true statement.

I now smile and say, "I know a truth that UTM will never utter." "Either UTM makes false statements, or it can not know the truth-value of all statements."

So UTM is not truly universal.

Q.E.D.