Russell's Paradox No More! Good news! I was just driving around after having done some posting and I realized something. I know the solution to Russell s paradox! I only just. http://www.toequest.com/forum/noumena/1314-russells-paradox-no-more.html
Abstract Math: Sets: Russell's Paradox Apr 15, 2008 Consequences of Russell s Paradox. We now know that setbuilder notation can t be depended on to give a set. To get around this difficulty, http://www.abstractmath.org/MM/MMRussellsParadox.htm
Extractions: MPBodyInit('MMRussellsParadox_files') abstractmath.org help with abstract math Produced by Charles Wells. Home Website Contents Website Index Posted 15 April 2008 The setbuilder notation has a bug: for some assertions P(x), the notation MPSetEqnAttrs('eq0001','',3,[[46,15,5,-1,-1],[58,19,6,-1,-1],[75,24,7,-1,-1],[65,22,7,-1,-1],[87,28,9,-1,-1],[110,35,11,-1,-1],[183,57,18,-2,-2]]) does not define a set MPSetEqnAttrs('eq0001','',3,[[46,15,5,-1,-1],[58,19,6,-1,-1],[75,24,7,-1,-1],[65,22,7,-1,-1],[87,28,9,-1,-1],[110,35,11,-1,-1],[183,57,18,-2,-2]]); MPNNCalcTopLeft(document.mpeq0001ph,'') MPDeleteCode('eq0001') Let P(x) be the assertion "x is a set ". Then if MPSetEqnAttrs('eq0002','',3,[[69,15,5,-1,-1],[89,19,6,-1,-1],[113,24,7,-1,-1],[100,22,7,-1,-1],[133,28,9,-1,-1],[168,35,11,-1,-1],[279,57,18,-2,-2]]) were a set , it would be the set of all set s . However, there is no such thing as the set of all sets . This can be proved using the theory of infinite cardinals, but not here . (See the discussion following Theorem 5 in Suber’s presentation of set theory MPSetEqnAttrs('eq0002','',3,[[69,15,5,-1,-1],[89,19,6,-1,-1],[113,24,7,-1,-1],[100,22,7,-1,-1],[133,28,9,-1,-1],[168,35,11,-1,-1],[279,57,18,-2,-2]]);
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Extractions: W ARNING - This does not shut off at the neck-tying so you will see Saddam dropping. The judge who sentenced Saddam Hussein to die within 30 days certain could have made it more interesting if he added this fillip: "And it will be on a day that you do not expect it." As to why this could be fun, a little background first. In 1966 I was sitting in a class at the Hebrew University in Jerusalem studying the Foundations of Mathematics taught entirely in Hebrew (which language I had just started learning the year before). It was the most fascinating course I had ever studied and the most revealing about the very structure and nature of existence itself: that mathematical objects exist independently of human observation and that rather than being the creators of these objects we humans merely discover them. It was during this time that I came across Russell's Paradox and other seeming contradictions. Here is the Unexpected Hanging Paradox: A man condemned to be hanged was sentenced to die within 30 days. The prisoner was told by the judge that he would be hanged at noon on one of the next 30 days and on a day that he would not be expecting it. The judge promised that this execution would be followed exactly to these two conditions.
