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Problems with definitions of what a set could be go back very far. Note that since George Boole (1815 - 1864) showed how Set Theory and Logic are really the same thing, these problems can crop up in a logical background. The first to spot a difficulty was the Greek philosopher Epimenedes in the 6th Century BC. He said, *All Cretans are liars* but since he was himself a Cretan one may deduce that if he was telling the truth, he was lying and vice versa. This attracted a lot of debate and became well-known enough to be referred to by St Paul (in his Epistle to Titus). It was made into a clearer paradox by the 4th Century BC philosopher Eubulides who said, *This statement is false* which leads to the same kind of problem.

It might seem that these problems of "self-reference" are not too serious, but when in the 19th Century mathematicians tried to define sets of elements by statements like *X* = { *x* | *x* satisfies some property *P*}, this was exactly the problem they ran into. One difficulty is that sets can actually contain sets as members. For example, the set {{1}, {1, 2}} is a perfectly satisfactory set. It is even possible for a set to contain itself as a member.

For example, let *A* be the set of all sets referred to in this page. Then *A* ∈ *A*.

The Welsh-born mathematician and philosopher Bertrand Russell (1872 - 1970) noticed that if one defined the set *R* = the set of all sets which do not contain themselves, if *R* ∈ *R* then *R* ∉ *R* and vice versa.

The German mathematician Gottlob Frege (1848 - 1925) was almost at the end of a major treatise on the foundations of arithmetic when Russell sent him this example. He added an acknowledgement to his treatise:

A scientist can hardly meet with anything more undesirable than to have the foundation give way just as the work is finished. In this position I was put by a letter from Mr Bertrand Russell as the work was nearly through the press.

Frege modified his axiom system to take account of this paradox, but in fact the system that he produced was then inconsistent. Russell himself spent many years working with Alfred Whitehead (1861 - 1947) to produce a three volume work *Principia Mathematica* (named after

There are several axiomatic systems for Set Theory. Among them are NBG (von Neumann-Bernays-Gödel) and ZSF (Zermelo-Skolem-Fraenkel). Later mathematicians, Kurt Gödel (1906 - 1978) in particular, showed that any axiomatic system for set theory cannot be wholly satisfactory. In any such system there are propositions that cannot be proved or disproved within the axioms of the system. In particular the consistency of the axioms cannot be proved.

You may feel, like Henri Poincaré (1854 - 1912)

Later generations will regard Mengenlehre(set theory)as a disease from which one has recovered.

(Actually, whether or not he actually said this is a matter of debate among historians of mathematics. See *The Mathematical Intelligencer* **13** (1991)).

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