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kpknudson.com | ||
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thatsmaths.com
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| | | | | In last week's post, we defined an extension of parity from the integers to the rational numbers. Three parity classes were found --- even, odd and none. This week, we show that, with an appropriate ordering or enumeration of the rationals, the three classes are not only equinumerate (having the same cardinality) but of equal... | |
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xorshammer.com
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| | | | | There are a number of applications of logic to ordinary mathematics, with the most coming from (I believe) model theory. One of the easiest and most striking that I know is called Ax's Theorem. Ax's Theorem: For all polynomial functions $latex f\colon \mathbb{C}^n\to \mathbb{C}^n$, if $latex f$ is injective, then $latex f$ is surjective. Very... | |
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www.jeremykun.com
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| | | | | Problem: Show there are finitely many primes. "Solution": Suppose to the contrary there are infinitely many primes. Let $ P$ be the set of primes, and $ S$ the set of square-free natural numbers (numbers whose prime factorization has no repeated factors). To each square-free number $ n \in S$ there corresponds a subset of primes, specifically the primes which make up $ n$'s prime factorization. Similarly, any subset $ Q \subset P$ of primes corresponds to a number in $ S$, since we can simply multiply all numbers in $ Q$ together to get a square-free number. | |
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unstableontology.com
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| | | This is an account of free choice in a physical universe. It is very much relevant to decision theory and philosophy of science. It is largely metaphysical, in terms of taking certain things to be basically real and examining what can be defined in terms of these things. The starting point of this account is... | ||
