“Every morning I would sit down before a blank sheet of paper. Throughout the day, with a brief interval for lunch, I would stare at the blank sheet. Often when evening came it was still empty… . [T]he two summers of 1903 and 1904 remain in my mind as a period of complete intellectual deadlock… . [I]t seemed quite likely that the whole of the rest of my life might be consumed in looking at that blank sheet of paper.”
That is from Bertrand Russell’s autobiography. What was stumping him was the attempt to find a definition of “number” in terms of pure logic. What does “three,” for example, actually mean? The German logician Gottlob Frege had come up with an answer: “three” is merely the set of all threesomes, the set of all those sets whose members can be exhaustively paired off with Larry, Curly, and Moe.
Most of the great truths of our time are best expressed as probabilities. This has taken some getting used to—most of us, in fact, are still not used to it.
However, if the concept “the set of all sets with a certain property” can be used indiscriminately, as Frege used it, then we can construct the set wof all sets that are not members of themselves. The set of all turtles is not a member of itself, since it is a set, not a turtle.
