vol 4: Glossary
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Russell's paradox
[Hazewinkel, Russell's paradox] [Borowski, Russell's paradox]
An antinomy, contradiction or paradox is a situation in which two mutually contradictory statements (p and not-p) are demonstrated, each one having been deduced by means that are convincing from the point of view of the same theory.
Russell worked on the paradox problem for years. He once wrote:
'Every morning', he later wrote, 'I would sit down before a blank sheet of paper. Throughout the day, with a brief interval for lunch, I would stare at the blank sheet. Often when evening came it was still empty. ... It seemed quite likely that the whole of the rest of my life might be consumed in looking at tha tblank sheet of paper.' Monk, 168
he set of all sets X which are equinumerous with Y (ie for which there is a one-one correspondence between Y and X ). We define card Y =< card Z to mean that Y is equinumerous with a subset of Z; by card Y < card Z we mean card Y =< card Z and card Y is not equal to card Z.
Cantor proved that if P(Y) is the set of all subsets of Y, then card Y < card P(Y). (Cantor's Theorem)
Let C be the universal set, ie the set of all sets. Now P(C) is a subset of C, so it follows that card P(C) =< card C .
On the other hand, by Cantor's theorem, card C < card P(C).
The Schroeder-Bersten Theorem asserts that if card Y =< card Z and card Z =< card Y, then card Y = card Z. Hence card C = card P(C), contradicting Card C < card P(C). Mendelson, 2.
The existence of russell's paradox points to a weakness in the assumptions made in the derivation of the paradox. In particular, the assumption of an all inclusive (universal) set seems suspect. As Cantor himself proved, there appears to be no largest set.
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