Page 8

		the hypothesis "all positive integers except 34." Suppose that the list has reached 61 at this point. Then continue with 34, 63, 64, 65, 66, 67, . . . until you hear "all positive integers except 62." If you continue in this devious way, one of two things will happen. Either:

		(a) your friend will go for the bait each time, and thereby change her hypothesis infinitely often, or

		(b) she will at some point refuse to adopt the conjecture that you intended for her.

		In both cases your friend will fail to make a last, correct conjecture about the list you have made. Moreover, in both cases the list you make belongs to the game. To see this, consider the two cases. In case (a), you will end up listing every positive integer. Since this set is a member of the initial collection C, your list represents a legitimate choice for Nature at the start of the game. In case (b), you will end up listing some set consisting of every positive integer with a sole exception. This set is also in C. Thus, in both cases your friend's guessing rule fails on some list for a set that might have been Nature's initial choice. Hence her guessing rule is not guaranteed to win the new game, which proves 1.4. (A more rigorous version of the proof is given in Chapter 3.)

		Let's play the last game again (with the extended collection C), but this time within a slightly different paradigm. Instead of being able to arbitrarily order the chosen set, Nature is now required to present the set in increasing order. So there is just one possible listing of any given set in C. For this paradigm, it is easy to formulate a guessing rule that wins in all cases (try stating such a rule).

		The foregoing variations point to a basic question about any, well-defined paradigm. The question is: For what collections of realities can winning guessing rules be formulated? This question is a dominant theme of our book.

		§1.4 Discussion of the Paradigms

		We now comment on various aspects of the paradigms just introduced. In fact, our remarks are relevant to almost all of the paradigms discussed in this book.

		§1.4.1 Possible Realities as Sets of Numbers

		Limiting the possible realities to sets of positive integers is not as austere as it might seem at first. This is because integers may be conceived as codes for objects and events found in scientific or developmental contexts. For example, the sentences to which children