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Using Heft 15 [volume 15] of Gödel's still-unpublished Arbeitshefte [working notebooks], John W. Dawson Jr. conjectures that Gödel discovered a proof for the independence of the axiom of choice from finite type theory, a weakened form of set theory, while in Blue Hill in 1942.
43 = (−9) × (−5) + (−2) and −2 is the least absolute remainder. In the division of 42 by 5, we have: 42 = 8 × 5 + 2, and since 2 < 5/2, 2 is both the least positive remainder and the least absolute remainder. In these examples, the (negative) least absolute remainder is obtained from the least positive remainder by subtracting 5 ...
Without ko, Go is PSPACE-hard. [2] This is proved by reducing True Quantified Boolean Formula , which is known to be PSPACE-complete, to generalized geography , to planar generalized geography, to planar generalized geography with maximum degree 3 , finally to Go positions.
Kurt Gödel Collected works, Vol. I. Oxford University Press, pp. 144–195. ISBN 978-0195147209. The original German with a facing English translation, preceded by an introductory note by Stephen Cole Kleene. —, 1951, "Some basic theorems on the foundations of mathematics and their implications", in Solomon Feferman, ed., 1995.
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Life and death (死活) is a fundamental concept in the game of Go, where the status of a specific group of stones is determined as either being "alive", where they may remain on the board indefinitely, or "dead", where the group will be "captured" and removed from the board. The basic idea can be summarized by: