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Erasing Russell's Paradox

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Cantor�s Theorem states that �every set is strictly dominated by its power set, or, in other words, |X| < |P(X)| for all X.� (Halmos p. 93) The usual proof of this theorem leads to a paradox similar to Russell�s Paradox: see http://www.u.arizona.edu/~miller/finalreport/node3.html

The reason for the paradox is simply that the theorem does not hold. A previous version of this paper stated that �based upon my paper entitled, �Erasing Russell�s Paradox�, Cantor's Theorem must hold. Axiom 3 from this paper states, �All non-empty sets can be ungrouped into primary objects� . Because an empty set contains no members it therefore cannot be ungrouped.� That assertion is contrary to the very premise that the referenced paper is based upon � sets are not objects. Therefore, the power set is not a valid set definition and, if X is any non-empty set, P(X) ~ X.

Based upon my paper, �Erasing Russell�s Paradox�, Axiom 3, we know that all members of P(X) can be ungrouped into primary objects. By the definition of subset, we know that these primary objects must be members of X and nothing more than members of X. The empty set is not a primary object. A set is defined to be a grouping of primary objects (axiom 2 from my paper on Russell�s Paradox). If set were a primary object, then its definition would be circular. Thus the empty set is not an object and cannot be ungrouped into an object. So when ungrouping a set of subsets, the empty set must be discarded. Therefore, for all X, P(X) ~ X, which means Cantor�s Theorem cannot hold.

� Copyright 2001, 2002 David B. Lowe

Last Modified: December 2, 2002

REFERENCES

Halmos, Paul R. Naive Set Theory. New York: Springer-Verlag, 1974.