Page 6

		uniquely described by the expression "all positive, even integers." The describable sets are the theoretically possible realities of the current paradigm (in the sense of 1.1a).

		To play the game, it will help to focus on a proper subset of all these realities, namely, the subcollection C defined as follows. C contains all sets that consist of every positive integer with a sole exception. Plainly, every set in C is describable; the set {1, 3, 4, 5, 6, . . .}, for example, is uniquely described by "all positive integers except for 2."

		In what follows, we shall play the role of Nature; you play the role of scientist. In our role as Nature, we select one member of C, and you (in your role as scientist) must discover the set that we have in mind. Clues about our choice will be provided in the following way. First, we shall order all the elements of the set in the form of a list; then the list will be presented one element at a time. There is no constraint on the list made from the chosen set, except that it must contain all the elements of the set, and only these. For example, one list of the set {2, 3, 4, 5, 6, 7, 8, 9, . . .} is: 3, 2, 5, 4, 7, 6, 9, 8, . . .. Aside from seeing the list's members presented one by one, you are provided no further information about it. A list of our set corresponds to 1.1c, the data made available about the possible reality chosen to be actual.

		Each time a number is presented, you may announce a conjecture about the set chosen from C at the beginning of the game (guesses about how we listed the chosen set are not required). Your guesses must take the form of English expressions that uniquely describe a set of positive integers. It is these expressions that constitute the intelligible hypotheses of our paradigm (see 1.1b). Your conjectures at any given moment will be based exclusively on the data available to you, so for purposes of this game you may be construed as a system that translates data into hypotheses. Indeed, any such system is considered to be a "scientist" within the current paradigm, in the sense of 1.1d.

		All of items 1.1a-d have now been specified. As for 1.1e, we stipulate that you win the game just in case you make only a finite number of conjectures, and the last one is correct.

		Let's play. We have selected a set and ordered it. Here is the first member of the list: 1. Guess, if you like. Next member: 3. Guess again, if you like. To abbreviate, here are the next ten members of the list: 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. Perhaps your latest conjecture is "all positive integers except for 2." That is a reasonable conjecture. However, it is wrong since according to our list the next number is 2. So go ahead and guess again. Here are the subsequent ten members: 15, 16, 17, 18, 19, 20, 21, 22, 23, 24. Perhaps now your latest conjecture is "all positive integers except for 14."

		The game goes on forever, so we interrupt it at this point to consider the paradigm