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homotopytypetheory.org | ||
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xorshammer.com
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| | | | | There is a class of all cardinalities $latex \mathbf{Card}$, and it has elements $latex 0$, $latex 1$ and operations $latex +$, $latex \cdot$, and so forth defined on it. Furthermore, there is a map $latex \mathrm{card}\colon\mathbf{Set}\to\mathbf{Card}$ which takes sets to cardinalities such that $latex \mathrm{card}(A\times B)=\mathrm{card}(A)\cdot\mathrm{card}(B)$ (and so on). Ordinary generating functions can be thought... | |
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unstableontology.com
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| | | | | (note: some readers may find the LaTeX more readable on LessWrong.) In this post I prove a variant of Gödel's completeness theorem. My intention has been to really understand the theorem, so that I am not simply shuffling symbols around, but am actually understanding why it is true. I hope it is helpful for at... | |
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ncatlab.org
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nhigham.com
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| | | The trace of an $latex n\times n$ matrix is the sum of its diagonal elements: $latex \mathrm{trace}(A) = \sum_{i=1}^n a_{ii}$. The trace is linear, that is, $latex \mathrm{trace}(A+B) = \mathrm{trace}(A) + \mathrm{trace}(B)$, and $latex \mathrm{trace}(A) = \mathrm{trace}(A^T)$. A key fact is that the trace is also the sum of the eigenvalues. The proof is by... | ||
