Question

1) Come up with your own story to illustrate Russel’s paradox informally.

Answer 1

Russell's paradox is based on examples like this: Consider a group of barbers who shave only those men who do not shave themselves. Suppose there is a barber in this collection who does not shave himself; then by the definition of the collection, he must shave himself. But no barber in the collection can shave himself. (If so, he would be a man who does shave men who shave themselves.)

Russell's paradox, which he published in Principles of Mathematics in 1903, demonstrated a fundamental limitation of such a system. In modern terms, this sort of system is best described in terms of sets, using so-called set-builder notation. For example, we can describe the collection of numbers 4, 5 and 6 by saying that x is the collection of integers, represented by n, that are greater than 3 and less than 7. We write this description of the set formally as x = { n: n is an integer and 3 < n < 7} . The objects in the set don't have to be numbers. We might let y ={x: x is a male resident of the United States }.

Seemingly, any description of x could fill the space after the colon. But Russell (and independently, Ernst Zermelo) noticed that x = {a: a is not in a} leads to a contradiction in the same way as the description of the collection of barbers. Is x itself in the set x? Either answer leads to a contradiction.

When Russell discovered this paradox, Frege immediately saw that it had a devastating effect on his system. Even so, he was unable to resolve it, and there have been many attempts in the last century to avoid it.