A while back I wrote an article discussing the question of whether you can find your phone number within pi. Just to summarize real quick, we do not currently have a proof that pi is a “normal number” and thus cannot definitively say that pi contains all finite sequences. So the only way to currently prove that your phone number exists within the digits of pi is to brute force it and look until you find it. 

I realized this is related to the halting problem. You can imagine a program that attempts to find your phone number within the digits of pi. It will either succeed and terminate, or it will run forever searching for an answer that doesn’t exist. But then I thought about it and realized something that bugged me. Doesn’t the halting problem only prove that paradoxical programs cannot be solved through a halt-determining algorithm? There is nothing contradictory or paradoxical related to the number pi. 

The halting problem exists in the same vein as paradoxes like the barber’s paradox, Russell’s paradox, liar’s paradox… etc blah blah. Basically you apply a condition or rule that contradicts itself making the condition impossible. But can’t you simply propose a version of the halting program that excludes contradictions? The halting problem only exists as a proofn’t (I think proof by contradiction should be called “proofn’t”) for a universal solution to all programs. That would change if the premise was all possible programs instead of just all programs. Given that contradictory programs are inherently impossible, we no longer have a proof that a useful halting program doesn’t exist. 

It’s like the question of “can an omnipotent god create a burrito so hot that even he cannot eat it?” I simply remove the omnipotent part out of the statement and suddenly this becomes a valid question. Granted we still don’t know the answer, but it is no longer an impossible question, just one we can’t currently answer. This brings me back to my idea of trying to find every phone number in pi. What if I was not trying to prove that pi is a normal number but merely trying to prove whether it was possible in the first place to determine that pi is a normal number at all? There’s like a succession of proof derivatives where you can always ask the question of “can you prove that you cannot/can prove any given thing?”