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Topic Math: Pinning it here!
« on: 2016-02-26 22:08:18 » |
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http://nautil.us/issue/2/uncertainty/the-deepest-uncertainty
[...]Correspondences can be used to compare the sizes of much larger collections than six goats—including infinite collections. The rule is that, if a correspondence exists between two collections, then they have the same size. If not, then one must be bigger. For example, the collection of all natural numbers {1,2,3,4,…} contains the collection of all multiples of five {5,10,15,20,…}. At first glance, this seems to indicate that the collection of natural numbers is larger than the collection of multiples of five. But in fact they are equal in size: every natural number can be paired uniquely with a multiple of five such that no number in either collection remains unpaired. One such correspondence would involve the number 1 pairing with 5, 2 with 10, and so on.
If we repeat this exercise to compare “real” numbers (these include whole numbers, fractions, decimals, and irrational numbers) with natural numbers, we find that the collection of real numbers is larger. In other words, it can be proven that a correspondence cannot exist between the two collections.
The Continuum Hypothesis states that there is no infinite collection of real numbers larger than the collection of natural numbers, but smaller than the collection of all real numbers. Cantor was convinced, but could never quite prove it.
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