Page 10

		of the theory, although we admit to having no proof of this claim.

		Another concession to mathematical simplicity can be noted here. Starting in the next chapter, the natural numbers {0, 1, 2, 3, . . .} will be used to construct possible realities, rather than the positive integers {1, 2, 3, 4, . . .}. This choice facilitates the use of techniques and results from the theory of recursive functions.

		§1.4.2 Intelligible Hypotheses

		We take hypotheses to be symbolic representations of a real or fictitious world. For example, most hypotheses announced in scientific journals are written in the symbols of the Roman alphabet, supplemented with mathematical notation. Alternatively, the alphabet might be drawn from some system of neural notation used by the brain to represent the structure of the ambient language.

		To be intelligible, a symbolic system must provide finite representations of the reality it is designed to depict, even if that reality is infinite in size. For example, the English expressions like "all positive integers except 5" is a finite string of letters that uniquely describes an infinite set. Computer programs can also be conceived as finite descriptions of sets of numbers. Specifically, program P can be taken as specifying the set of all numbers n such that P given input n eventually stops running. This is the approach described in the next chapter and used throughout the sequel. The emphasis on computer programs as hypotheses stems in part from the desire for technological applications. Moreover, it is felt that programs stand in a particularly intimate relation to the sets they describe, inasmuch as they provide a means for recognizing the members of the set. In contrast, the English description "all positive integers that Gauss ever wrote down" uniquely describes a set of numbers, but gives little access to its members.

		§1.4.3 Scientists

		In our sample paradigm above, scientists were conceived as systems that convert finite sequences of numbers into hypotheses. The scientist may thus be pictured as traveling down an infinite list of numbers, examining the finite amount of data available at any point in the voyage, and emitting hypotheses from time to time about the contents of the entire list, including the infinite, unseen portion. For most of the book, it will be assumed that scientists are mechanical, that is, simulable by a computer. Indeed, we shall usually equate scientists with computer programs. It will sometimes prove helpful, however, to remove the assumption of computability from our conception of scientists, in which case