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Originally posted by drew hempel
reply to post by garritynet
Philosophy (1999), 74:2:169-176 Cambridge University Press
Copyright © The Royal Institute of Philosophy 1999
Did the Greeks Discover the Irrationals?
Philip Hugly and Charles Sayward
A popular view is that the great discovery of Pythagoras was that there are irrational numbers, e.g., the positive square root of two. Against this it is argued that mathematics and geometry, together with their applications, do not show that there are irrational numbers or compel assent to that proposition.
So then the problem is that if you believe that for any infinite set there's a set that's even larger, what happens if you apply this to the universal set, the set of everything? The problem is that by definition the set of everything has everything, and this method supposedly would give you a larger set, which is the set of all subsets of everything. So there's got to be a problem, and the problem was noticed by Bertrand Russell. Bertrand Russell Cantor I think may have noticed it, but Bertrand Russell went around telling everyone about it, giving the bad news to everyone! --- At least Gödel attributes to Russell the recognition that there was a serious crisis. The disaster that Russell noticed in this proof of Cantor's was the set of all sets that are not members of themselves, that turns out to be the key step in the proof. And the set of all sets that aren't members of themselves sounds like a reasonable way to define a set, but if you ask if it's inside itself or not, whatever you assume you get the opposite, it's a contradiction, it's like saying this statement is false. The set of all sets that are not members of themselves is contained in itself if and only if it's not contained in itself.
The simplicity of the discovery makes it "a beautiful result," Nemenman says. "We hope that this theoretical finding will also have practical applications."