Word: godell
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Soon it dawned on a few insightful souls, Godel foremost among them, that this way of looking at things opened up a brand-new branch of mathematics--namely, metamathematics. The familiar methods of mathematical analysis could be brought to bear on the very pattern-sprouting processes that formed the essence of formal systems--of which mathematics itself was supposed to be the primary example. Thus mathematics twists back on itself, like a self-eating snake...
Bizarre consequences, Godel showed, come from focusing the lens of mathematics on mathematics itself. One way to make this concrete is to imagine that on some far planet (Mars, let's say) all the symbols used to write math books happen--by some amazing coincidence--to look like our numerals 0 through 9. Thus when Martians discuss in their textbooks a certain famous discovery that we on Earth attribute to Euclid and that we would express as follows: "There are infinitely many prime numbers," what they write down turns out to look like this: "84453298445087 87863070005766619463864545067111." To us it looks...
...such a simple shift of perspective, Godel wrought deep magic. The Godelian trick is to imagine studying what might be called "Martian-producible numbers" (those numbers that are in fact theorems in the Martian textbooks), and to ask questions such as, "Is or is not the number 8030974 Martian-producible (M.P., for short)?" This question means, Will the statement '8030974' ever turn up in a Martian textbook...
...Godel, in thinking very carefully about this rather surreal scenario, soon realized that the property of being M.P. was not all that different from such familiar notions as "prime number," "odd number" and so forth. Thus earthbound number theorists could, with their standard tools, tackle such questions as, "Which numbers are M.P. numbers, and which are not?" for example, or "Are there infinitely many non-M.P. numbers?" Advanced math textbooks--on Earth, and in principle on Mars as well--might have whole chapters about M.P. numbers...
...thus, in one of the keenest insights in the history of mathematics, Godel devised a remarkable statement that said simply, "X is not an M.P. number" where X is the exact number we read when the statement "X is not an M.P. number" is translated into Martian math notation. Think about this for a little while until you get it. Translated into Martian notation, the statement "X is not an M.P. number" will look to us like just some huge string of digits--a very big numeral. But that string of Martian writing is our numeral for the number...