Mathematics begins with an injunction - make a distinction.
And ends with being unable to decide whether a problem is solvable or not.
It's hard to believe I know, but mathematics has a beginning and an end.
I have been touting mathematics as the language of God, but I was wrong.
God, as we all know, is dead but some of us hoped we could hear his last whispers in mathematics. But no, mathematics itself teaches us that mathematics cannot describe the world.
So we are left in the Universe, bereft children without God or logic. All we have left is ourselves.
But alas the lust for Certainty remains in fundamentalism. Although we now know with mathematical certainty that the foundations are air.
And as you note, this is a paradox that has driven four of our greatest mathematicians to madness and suicide, Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing.