But of what use is a “blog” if it’s not worth reading. Therefore I’m changing that, with questions as advertised, and some dry humor on top.
Firstly, the big question. Why make a blog? What do I gain as a being from typing my endless ideas into the vastness of space?
An interesting question for sure, but can that question be answered? Here’s a tangent on that very subject:
Godel’s Incompleteness Theorem. It’s a very complicated and heavy-weight mathematics proof showing something that shocked science, logic, and the academic world at the time. It seemingly comes up with a proof of the impossible: That logic itself can never be complete.
The core of this proof is as follows, without all the jargon and messy mathematics that would scare off the average reader. Godel proved that any system using Axioms (statements that are true no matter what, like X is equal to itself), will never be able to have enough axioms to prove everything. In other words, at some point you will need new rules of logic to solve new problems.
He did this in a very clever (albeit paradoxical and unconventional) way. First, he starts with any old system using Axioms of logic. Then, he poses a seemingly trick problem:
“This statement is NOT provable by the current axioms”
This question (because of logic and other reasons) can only have 2 answers: Either it’s a true statement, or it’s false. If it’s false, an immediate paradox arises: If it’s false, we had a proof for it being false. And if we had a proof from the axioms that it’s false, that also means it was TRUE! This is a contradiction, which means that it can’t possibly be false, right? This leaves us with only 1 other option. It must be true! However, this also leads to a problem: How did we ever prove it was true? If the statement is true, it’s also impossible to prove! Therefore, this statement is both TRUE and UN-PROVABLE.
But hold on a second, didn’t we just prove it’s correct? Doesn’t that mean that the statement must also be false? Not entirely. You see, we DID prove that this statement is correct, but we had to add a few new axioms to do it. We had to add the axiom that “this statement is NOT provable by the current axioms” to our list of axioms to make it work. It’s complicated, but without diving into the extremely complicated proof itself I can’t show you why this is the case.
So we solved it. This statement is unprovable, and also must be correct. We’re done, right? Here’s the genius of Godel’s proof: We can just ask the same question again! “This OTHER statement is NOT provable by the current axioms”, proving that there are an INFINITE number of unknown and unexplored axioms, meaning that logic will forever rest uncompleted.
Overall, this whole process might seem a little pointless. Why do we bother asking these questions about a certain statement being provable? Maybe it just applies to these odd, self-contradicting sentences. This, at least, was mathematician’s hope, until this too was shattered. A brilliant paper called the Paris – Harrington Theorem came along and proved, using a test model of a potential statement that needed to be proven, that certain problems in mathematics, science and logic could contain these dangerous “Godellian” sections that ultimately mean that they can’t and will never be proven by the axioms. Fear hung over everyone as debates raged over the possibility of very high-level problems being potentially unprovable, like the Twin Primes conjecture, Goldbach’s conjecture, and the Riemann Hypothesis.
Thankfully, we can expand the axioms a bit to help us with these particular examples. If we can manage to prove that these problems cannot be proven, then use our new axioms created by Godel like “statements that you can prove cannot be proven by the axioms are true”, we could potentially have a workaround to proving these great mysteries.
This is the first of hopefully many talks and ideas thrown around by me.
I’d love your feedback. Ask me anything, really. Contradict me.