Gödel’s incompleteness theorem is well known for proving that the dream of most mathematicians to formulate foundations for a complete and self-consistent theory of mathematics is a futile exercise.
Gödel proved that any set of axioms you could posit as a possible foundation for math will inevitably be incomplete; there will always be true facts about numbers that cannot be proved by those axioms. He also showed that no candidate set of axioms can ever prove its own consistency (1).
In essence, the incompleteness syllogism by Gödel starts from talking abour logical propositions (or mathematical propositions if you like) and ends up with a proposition that talks about the validity of… itself. This proposition which we might as well call reads something like “I cannot be proved”.
This leads to a dead-end.
If it can be proved, then it means that it cannot.
And vice versa.
So it is essentially a logically true proposition (since indeed it cannot be proved) but which cannot actually be proved within the axiomatic system at hand.
Hence, the incompleteness.
Essentially this is something the ancient Greeks have thought of thousands of years ago; something which they formulated in the famous Epimenides paradox. Epimenides was a man from Crete who said the following simple thing: “All Cretana are liars”.
Well, this ends up in the same dead-end as the proposition mentioned above. If Epimenides is truthful, then he is a liar since he is Cretan and all Cretes are liars. If he is a liar, then he is telling the truth! And, thus, he is a liar!
A self-reference paradox which essentially destroys the hope of mathematicians around the world for a consistent and full way to formulate mathematics. It is weird, but also important to mention here, that self-reference is the basis of our existence. Consciousness, our ability to speak about our self and our own existence and being, is the foundation of our essence as human beings. Without that, we would be nothing than complex machines.
But how can this dead-end be surpassed or perhaps by-passed?
Well, it cannot actually.
You ctu right through it.
I was in a discussion the other day about the above topics and when the Epimenides paradox was mentioned, one immediate reaction that I got was the simple “So the solution is that he is not from Crete” (!)
What?! I answered. But I told you he was a Cretan.
Sure. He was.
What is he wasn’t?
Then there wouldn’t be any paradox!
In the same sense…
What if the logical proposition…
“I am false”
is not a… proposition?
Then all problems would be solved!
But if it is not a logical or mathematical proposition then what is it? Well, as I said above, self-reference is not mathematics per se. It is more of a metaphysical reference to existence and being. A proposition talking about… itself is no more a proposition but an attempt to speak with the abyss. It is more God talking to humans than humans trying to talk with God. Such a thing could be many things, but ‘simply’ a logical (mathematical) proposition not.
But this is gibberish, one might counter-argue.
Sure, it can be.
(Gibberish like the Russel way out of his paradox?)
If you really think a Cretan would ever call himself a liar.
Sure, it can be.
If you accept that a proposion can ever referto itself.
But it cannot.
In a cosmos where only humans can talk for themselves.
In a cosmos where mathematics cannot prove themselves.
In a world where endless-loop paradoxes exist.
In a life which is full with nothing but them.
Paradoxes were the end of the hopes of mathematicians. They alone can be the ones which will instil hope in the once again.
Look around Cretan.
If you cannot prove that there is a sea…
Will you ever lie that you are swimming?