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rjlipton.com | ||
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xorshammer.com
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| | | | | Let $latex \mathrm{PA}$ be Peano Arithmetic. Gödel's Second Incompleteness Theorem says that no consistent theory $latex T$ extending $latex \mathrm{PA}$ can prove its own consistency. (I'll write $latex \mathrm{Con}(T)$ for the statement asserting $latex T$'s consistency; more on this later.) In particular, $latex \mathrm{PA} + \mathrm{Con}(\mathrm{PA})$ is stronger than $latex \mathrm{PA}$. But certainly, given that... | |
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jdh.hamkins.org
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| | | | | I'd like to share a simple proof I've discovered recently of a surprising fact: there is a universal algorithm, capable of computing any given function! Wait, what? What on earth do I ... | |
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dvt.name
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| | | | | In the previous blog post in this series, we looked at Gödel's First Incompleteness Theorem, and came to the amazing conclusion that we can't compute certain kinds of functions in formal systems (like Javascript). Specifically, we looked at a special function, , which turned out to be non-computable. In case we forgot, the first incompleteness ... | |
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thehighergeometer.wordpress.com
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| | | Here's a fun thing: if you want to generate a random finite $latex T_0$ space, instead select a random subset from $latex \mathbb{S}^n$, the $latex n$-fold power of the Sierpinski space $latex \mathbb{S}$, since every $latex T_0$ space embeds into some (arbitrary) product of copies of the Sierpinski space. (Recall that $latex \mathbb{S}$ has underlying... | ||
