# Russell's Paradox

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, |

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

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:

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*:

At the end of the Lent Term, Alys and I went back to Femhurst, where I set to work to write
out the logical deduction of mathematics which afterwards became
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 It may not be quite obvious but Russell's Paradox is just a variant of the diagonal argument. ## Reference- W. Dunham,
*The Mathematical Universe*, John Wiley & Sons, NY, 1994. - M. Gardner,
*aha! Gotcha. Paradoxes to puzzle and delight*, Freeman & Co, NY, 1982 - D. R. Hofstadter,
*Metamagical Themas*, Basic Books, Inc., 1985, Chapter 16. - J. A. Paulos,
*Beyond Numeracy*, Vintage Books, 1992 - Rudy Rucker,
*Infinity and the Mind*, Princeton University Press, Princeton, NJ, 1955
## Self-reference and apparent self-reference- Does It Blink?
- Apparent paradox
- Set of all subsets
- An Impossible Page
- Russell's paradox
- An Impossible Machine
- A theorem with an obvious proof
- The Diagonal Argument
- A link to a very similar puzzle
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