Poincaré disliked Peano's work on a formal language for mathematics, then called "logistic." He wrote of Russell's paradox, with evident satisfaction, "Logistic has finally proved that it is not sterile. At last it has given birth - to a contradiction." from R.Hersh, What is Mathematics, Really? Oxford University Press, 1997

Sets are defined by the unique properties of their elements. One may not mention sets and elements simultaneously, but one notion has no meaning without other. The widely used Peano's notation

A = {x: x has property P}

incorporates all the pertinent attributes: a set A, a property P, elements x. But, of course, one does not always use the formal notations. For example, it's quite acceptable to talk of the set of all students at the East Brunswick High or the set of fingers I use to type this sentence. The space being limited, some sets are described on this page and some are not. Let's call russell the set of all sets described on this page. Just driving the point in: russell's elements are sets described on this page. Note that this page is where you met russell. For it's where it was defined after all. So russell has an interesting property of being its own element: russell∈russell.

With the example of russell it's apparent that some sets contain themselves as elements while others do not. Let RUSSELL stand for the set of all sets that are not their own elements. What may be said about RUSSELL? Which is it?

Assuming RUSSELL∈RUSSELL leads to a contradiction for, by definition, RUSSELL does not contain itself. Assuming RUSSELL∉RUSSELL implies that RUSSELL satisfies the definition and, hence, RUSSELL∈RUSSELL. Impossibility.

That RUSSELL is such a set that neither RUSSELL∈RUSSELL nor RUSSELL∉RUSSELL has been discovered by Bertrand Russell (1872-1970) in 1901. This is how he described the event in his Autobiography:

Principia Mathematica is the book Russell wrote with Alfred North Whitehead where they gave a logical foundation of Mathematics by developing the Theory of Types that obviated the Russell's paradox. This assertion may become more convincing after a look at the page 362 of Principia Mathematica where Russell and Whitehead finally proved that 1 + 1 = 2.

There are a few more things that should be mentioned in this context. I'll leave the discussion for now. There might be a chance I'd return to this page after settling the question whether or not RUSSELL∈russell. (Of course, russell∉RUSSELL.) Meanwhile you may ponder a question concerning the set RuSSeLL of all sets not described on this page. Is it true that RuSSeLL∈russell?

It may not be quite obvious but Russell's Paradox is just a variant of the diagonal argument.

## Reference

1. W. Dunham, The Mathematical Universe, John Wiley & Sons, NY, 1994.
2. M. Gardner, aha! Gotcha. Paradoxes to puzzle and delight, Freeman & Co, NY, 1982
3. D. R. Hofstadter, Metamagical Themas, Basic Books, Inc., 1985, Chapter 16.
4. J. A. Paulos, Beyond Numeracy, Vintage Books, 1992
5. Rudy Rucker, Infinity and the Mind, Princeton University Press, Princeton, NJ, 1955