In Aha! Gotcha : Paradoxes to Puzzle and Delight Martin Gardner uses paradox to include "any result so contrary to common sense and intuition that it invokes an immediate emotion of surprise." He suggests that the four main types of paradox are thus:

1 An assertion which seems false but is actually true.
2 An assertion that seems true but is actually false.
3 A line of reasoning that seems impeccable but which leads to a logic contradiction. (More commonly called a fallacy.)
4 An assertion whose truth or falsity is undecidable.

CONTINUE

The importance of paradoxes should not be underestimated. On the one hand they may simply be a source of entertainment and fascination for the casual observer, but beyond that they play an important role in furthering our understanding of the world around us. They lure us in, infuriate us, and lead to lines of enquiry that might otherwise have been neglected.

CONTINUE

Gottlob Frege was a german mathematician who devoted many years of his life to a treatise on the foundations of arithmatic. His work contained frequent referrences to the class of all classes that have a given property. He was just about to publish the second volume (some 10 years after the publication of the first) when he received a communication from Bertand Russell. In this communication Russell outlined what is now know as Russell's Paradox.

FREGE'S RESPONSE

"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."

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