I've read a little bit about this theorem, and it's really weird and fascinating. Back in the early 1900's, Bertrand Russell and Alfred Whitehead created a system of arithmetic on which mathematics could be more firmly grounded. There was an ideal of being able to demonstrate every mathematical theorem so far accumulated in the discipline in a really rigorous fashion, building from the most fundamental pieces of set theory and logic. With a system as rigorous as the one Russell and Whitehead devised, it was hoped that every true mathematical statement could be thoroughly proved from first principles. Godel ruined the party and proved in 1931 that any sufficiently powerful system of arithmetic becomes capable of talking, not only about numbers and their properties, but about itself, which allows for paradox to creep in.
When a system becomes capable of talking about itself, it can say things like "this statement is not provable within the present system". Such a statement is true, but manifestly unprovable. If you were to find a proof for that statement, then the statement would be false, suggesting your proof contradicts the premise you're trying to demonstrate. On the other hand, if no proof exists for such a statement, then the statement is true. Godel's deathblow was to show that all similarly powerful systems of arithmetic will be capable of generating such true but unprovable statements. So the dream of being able to prove every true mathematical proposition came to an end. At least that's my understanding of it.
One question Godel has always prompted in my mind is, if truth is independent of provability, then what exactly does it mean for a mathematical statement to be true? If we say that X is true, do we just mean that no counterexamples to X exist?
Well I've had a few drinks tonight and so I'm properly equipped to talk about math. I often wonder whether mathematics is merely a human tool invented to aid understanding, or whether in some sense it's what really drives the universe. The success of mathematics in physics, to take only one example, is somewhat puzzling. Why should an abstract conceptual construct so precisely reflect the inner workings of the universe? Might it be the case that mathematics is in some way more fundamental than physical law itself? Of course these questions are unanswerable at this time, but Godel has something to offer. The incompleteness theorems show us that mathematics, when rigorously applied, allow the emergence of language and higher order meaning. Self reference strikes me as a very strange phenomenon, and the fact that it emerges inevitably from certain systems of arithmetic makes me think that it is much more than a mere tool. There is something deep and mysterious lurking in the fundamentals of mathematics. Do mathematicians invent theorems or discover them? I would say the latter. Few if any disciplines so completely remove the role of the observer as mathematics. Does that not elevate math to a higher level of objectivity and permanence?
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