Among his many accomplishments, Bertrand Russell (x-y), collaborated with the great Alfred North Whitehead (x-y) to complete two volumes of Principia Mathematica (not to be confused with Newton's work of the same name), purporting to DERIVE MATHEMATICS FROM LOGIC.

One of the logical "tools" Russell used was the Multiplicity Postulate, which is equivalent to The Axiom of Choice, part of the Dogma of Platonism.

Briefly, AC DECLARES THAT A PROTOTYPE OF EVERY SET OR PATTERN CAN BE SET FORTH. The standard way of doing so is to WELL-ORDER ANY SET AND CHOOSE SUCH A REPRESENTATIVE FROM THIS ORDERING. But we do not know how to WELL-ORDER some sets.

For example, George Cantor (x-y), founder of Set Theory, showed hou you would attempt to write down an ORDERING OF ALL REAL NUMBERS GREATER THAN ZERO BUT LESS THAN ONE -- "decimal numbers" of the form: 0.d_ij, where i is the INDEX of a NUMBER in that PROPOSED ORDERING, and j IS the INDEX of DIGIT in the EXPANSION OF A SINGLE NUMBER. Then, Cantor showed how you can alsways used such a proposed ordering to CONSTRUCT A NUMBER THAN CANNOT BE IN THAT ORDERING. (It's comparable to COUNTING: No matter how "far" you count, you can always use that number to find its successor. Hence, you can't find the last number.) Since such a construction (called a "diagonalization") is always possible, you CANNOT WELL-ORDER THE REAL NUMBERS IN SUCH A FASION. And, even now, no one knows how to WELL-ORDER THE REAL NUMBERS.

But THE AXIOM OF CHOICE SAYS IT CAN BE DONE -- WITHOUT SHOWING HOW.

To explain the problem, Russell came up with this amusing illustration.

Suppose an infinite number of PAIRS OF SHOES lie behind a curtain. You can reach a hand in to retrieve one or more. But you cannot look behind the curtain. Problem: TO CHOOSE A REPRESENTATIVE OF EVERY PAIR OF SHOES BEHIND THE CURTAIN.

Easy. Use the RULE (ALGORITHM) THAT YOU CONSIDER ONLY LEFT SHOES. You retrieve a shoe. If a LEFT SHOE, you keep it. If a RIGHT SHOE, you put it back. By thie RULE, you will collect ONLY ONE SHOE FROM EACH PAIR.

But -- says Russell, closing the trap -- what about socks? Only the MULTIPLICITY POSTULATE or AXIOM OF CHOICE assures us that this can be done.