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		are exposed in the course of language acquisition (like all sentences of human language) are complex structures involving phonetic, syntactic, and semantic levels of representation. Their complexity notwithstanding, it may nonetheless be possible to enumerate all possible sentences in a kind of alphabetical order in something like the way pairs, triples, or quadruples of integers can be enumerated.4 If the enumeration can be carried out by a computable process, then it yields a useful correspondence between sentences and integers, and the latter can be used as codes for the former. In this case, a set of integers corresponds to a language, namely, the language whose sentences are coded by the integers in the set.5

		Notice that a correspondence of this kind requires that the set of coded entities be denumerable, i.e., have the same cardinality as the integers serving as codes. It might be thought that the restriction to denumerable domains excludes scientific contexts bearing on physical quantities whose values are arbitrary real numbers. However, the rational numbers provide sufficient precision in scientific practice, and the rationals are a denumerable set. So integers can also be used to code many situations involving physical quantities.

		For mathematical simplicity, the possible realities figuring in this book are taken to be sets of numbers, or else functions from numbers to numbers. We limit ourselves thereby to studying scientific or developmental contexts in which the relevant objects of inquiry (like sentences) can be coded as integers. It is our belief that much insight into empirical investigation can be achieved within this limitation, a claim that the reader will ultimately have to evaluate for him- or herself.6

		There is an additional, noteworthy property of the sets and functions that play the role of possible realities in most of what follows. They are "computable" in the sense of being manipulable and recognizable by computer programs (this will be made precise in the next chapter). It is important to recognize that most sets of numbers and most numerical functions are not computable. In fact, from the point of view of their respective cardinalities, the computable functions stand in the same relation to the class of all numerical functions as do the integers to the real line. It follows that by limiting attention to possible realities of a computable nature our theory does not embrace every conceivable scientific situation (we return to this point in Chapter 3). Once again, we believe that this restriction leaves a large and important class of scientific contexts within the purview

		4 For exposition of this kind of enumeration, see Boolos and Jeffery [21, Chapter 1].

		5 For more extended discussion see Weihrauch [192, Chapter 3.3].

		6 Paradigms involving more expressive scientific languages are discussed in Martin and Osherson [127, 128].