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| Nothing bigger than infinity, right? Well, what about second infinity? From Quanta: Mathematicians Measure Infinities and Find They’re Equal  Two mathematicians have proved that two different infinities are equal in size, settling a long-standing question. Their proof rests on a surprising link between the sizes of infinities and the complexity of mathematical theories. When someone brightly proclaims, "Nothing is impossible!" I have two possible responses, depending on my mood: 1) Even in a vacuum, there exist electromagnetic fields, quantum virtual particles, etc., so yes, technically, it's impossible to achieve "nothing." 2) Okay, if nothing is impossible, please, go ahead and count to infinity. And if I'm in a really bad mood, there's a third: 3) "You are." But despite my #2 response, which demonstrates that there are things that are, in point of fact, not possible, mathematicians understand quite a bit about the very useful (but probably entirely abstract) concept of infinity, one bit being that the infinity of integers is of a lower order, aka smaller, than the infinity of real numbers. The problem was first identified over a century ago. At the time, mathematicians knew that “the real numbers are bigger than the natural numbers, but not how much bigger. Is it the next biggest size, or is there a size in between?” said Maryanthe Malliaris of the University of Chicago, co-author of the new work along with Saharon Shelah of the Hebrew University of Jerusalem and Rutgers University. Just so we're clear, in math, "natural numbers" means positive integers, and "real numbers" means integers, fractions, and non-repeating transcendental numbers like pi. From what little about this stuff that I understand, there is a greater infinity of real numbers just between the integers 1 and 2 than the entire infinity of integers. Oh, and to make your mind spin even more, the infinity of natural numbers is the same size as the infinity of integers, which is the same size as the infinity of even numbers, which is the same size as the infinity of multiples of 100, and the infinity of prime numbers, and so on. You don't have to take my word for it, but mathematicians have proven this shit rigorously, as the article goes on to explain using pretty basic concepts, all because of the work of a dude named Georg Cantor. Then: What Cantor couldn’t figure out was whether there exists an intermediate size of infinity — something between the size of the countable natural numbers and the uncountable real numbers. He guessed not, a conjecture now known as the continuum hypothesis. Now, I'm obviously not a mathematician, but this stuff fascinates me to the point where I've read entire books about it. One of them, as I recall, pointed out later work that showed that the continuum hypothesis cannot be proven or disproven using the framework of mathematics, which this article also nods to: In the 1960s, the mathematician Paul Cohen explained why. Cohen developed a method called “forcing” that demonstrated that the continuum hypothesis is independent of the axioms of mathematics — that is, it couldn’t be proved within the framework of set theory. (Cohen’s work complemented work by Kurt Gödel in 1940 that showed that the continuum hypothesis couldn’t be disproved within the usual axioms of mathematics.) The article continues in even more detail and, no, there's not a lot of actual math in it; it's mostly written in plain language, and the only problem I had reading it was keeping all the names straight. Because while I don't understand much of mathematics, I understand it better than I understand people. My real point in posting this, however, is to show that some things are, in fact, impossible. But what is possible is to know what's impossible. |
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