Bizarre consequences, Gödel 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 like one big 46-digit number. To Martians, however, it is not a number at all but a statement; indeed, to them it declares the infinitude of primes as transparently as that set of 34 letters constituting six words a few lines back does to you and me.

Now imagine that we wanted to talk about the general nature of all theorems of mathematics. If we look in the Martians' textbooks, all such theorems will look to our eyes like mere numbers. And so we might develop an elaborate theory about which numbers could turn up in Martian textbooks and which numbers would never turn up there. Of course we would not really be talking about numbers, but rather about strings of symbols that to us look like numbers. And yet, might it not be easier for us to forget about what these strings of symbols mean to the Martians and just to look at them as plain old numerals?

By such a simple shift of perspective, Gödel wrought deep magic. The Gšdelian 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?

Gšdel, 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.

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