Goedel’s incompleteness theorem

At the age of twenty-five the Czech mathematician Goedel published his Incompleteness Theorems. The first incompleteness theorem is the most beautiful mathematic statement I have read, although I must confess that my mathematical knowledge is limited and random at best. But certain ideas appeal to me, it is not even clear if this means that I understand them and the appeal consists of an erupting euphoria that makes itself master of my shallow understanding: eureka, I understand! Or if it means that I stubbornly resist the idea of not grasping the essence of the thought. But eitherway, the aesthetic experience is overwhelming and rare, certainly not a daily commodity for a working man, more common I know for a student or a man unburdened by the obligation of having to make a living.

“For any consistent formal theory that proves basic arithmetical truths, it is possible to construct an arithmetical statement that is true 1 but not provable in the theory. That is, any consistent theory of a certain expressive strength is incomplete.”

True, Epimenides wrote his well-known paradox “Cretans are always liars” more than 2500 years ago. And sure, I acknowledge that there is an arbitrary element in my taste, of course, my epiphany is not a unique invention, I am simply subdued to it, the pupil knows its master. Logic is such a powerful temptation to our thought, that since Descartes our whole conception of truth has been based on it. The notion of an entity existing outside the realm of our thinking was confined to the realm of religion and metaphysics, but Goedel opened up the doors of perception to the unthinkable, the illogic. Plato’s degrees of knowledge were an absolute idea of delusion. How close were dream and certainty in Platonic philosophy!

I squint while reading it over again, the form sets itself instantly, but yet it has no context. This is the purity of Hegelian aesthetics, an enduring pleasure free of interest. I gain nothing from this understanding, I have lost none.

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