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		in more general terms. Let us say that a "guessing rule" is a set of instructions for converting the clues received up to a given point into a conjecture about the chosen set. Your own guesses may well have been chosen according to some guessing rule, and you might take a moment to attempt to articulate it.

		Now consider the following guessing rule:

		1.2 Guessing rule: Suppose that S is the set of numbers that have been presented so far. Let m be the least positive integer that is not a member of S. (S must be finite, so such a number certainly exists.) Emit the conjecture "all positive integers except for m" unless this was your last conjecture (in which case make no conjecture at all).

		To illustrate, if the numbers presented so far were {4, 5, 8, 1}, then rule 1.2 would direct you to conjecture "all positive integers except for 2" (unless you had just made this conjecture, in which case you would not do it again). You should be able to convince yourself of the following fact:

		1.3 Fact: No matter which set was chosen from C at the start of the game, and no matter what list was made from that set, consistent application of guessing rule 1.2 is a winning strategy; that is, if you use rule 1.2, then you win in all cases.

		Now let us modify the game by adding the set of all positive integers (without exception) to the initial collection C. So our choice of set as "actual reality" is expanded to include one new possibility, namely, {1, 2, 3, 4, 5, . . .}. This changes matters quite a bit. For example, guessing rule 1.2 is no longer guaranteed to succeed at the game. Indeed, it is clear that, faced with any listing for the new set {1, 2, 3, 4, 5, . . .}, the rule changes its guess infinitely often, and hence never produces a last, accurate conjecture. A more significant fact is the following.

		1.4 Fact: No guessing rule is guaranteed to win the new game. That is, for every guessing rule R there is a set in the (expanded) collection C and some way to list the set such that R fails to produce a last, correct conjecture on the list.

		The techniques needed to prove Fact 1.4 will be presented in Chapter 3. You can grasp the matter intuitively, however, by playing the new game with a friend. This time you play the role of Nature, and try to defeat your opponent with the following tactic. Begin with the list 2, 3, 4, 5, 6, 7, . . ., extending it until your friend announces the hypothesis "all positive integers except 1." Suppose that your list must be extended to 33 for this to happen. Then continue your list with 1, 35, 36, 37, 38, 39, 40, . . . until you have extracted