Palo Alto Institute for Advanced Study
2007-12-18
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Palo Alto Institute for Advanced Study 2007-12-18
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FOR IMMEDIATE RELEASE Paradox... OversightsParadox is thousands of years old. One would think that all important paradoxes in classical mathematics had been at least discovered, but...2003 Jul 04 Paradox has been around a long time. Most paradoxes have historically related to the concept of infinity, and modern mathematics has even made such paradoxes acceptable, whereas before they were taken as inconsistency in the bad sense. Although one would think that all important paradoxes in classical mathematics have been found, at least those relating to real number theory, set theory and other mature branches of mathematics, this is not the case. Real number theory has yet to recognize the Vanishing Remainders Paradoxes. For example, it can be shown that the rational number 1/3 cannot be a real number if by that we mean that it cannot be successfully represented as an infinite decimal expansion. It can also be shown that the square root of two cannot be such a real number using logic just like the ancient Greeks used to demonstrate that the square root of 2 could not be a rational number. There are More Real Number Paradoxes, that go into detail concerning the consequences of this class of paradoxes for real number theory. Also presented is a rigorous treatment of the Bijection Permutation Paradox (an analytical approach to the Good Shepherd’s Paradox, Golfer’s Paradox and Banker’s Paradox) that if set theory allows transfinite sets to be bijected with proper subsets of themselves, then it follows that there must exist bijections between non-empty sets and the empty set. Set subtraction has its counterpart in bijections that can be permuted while maintaining bijectivity invariant. This paradox along with its analysis and resolution forms the basis for the resolution of the Vanishing Remainders Paradox and its companions. Directly or by way of the Mathematics or the section.
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