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		SEG. To implement these functions we could conceive of three types of Turing Machine, each with its own domain. The following strategy is simpler, however. We now fix two recursive isomorphisms, one between SEQ and N, the other between SEG and N. The countability of SEQ and SEG (and their simple structure) guarantees the existence of these isomorphisms, whose details will not concern us. The isomorphisms allow the members of SEQ and SEG to be coded as natural numbers, and using these codes the same Turing Machine TM may be conceived alternately as computing any of the three kinds of functions distinguished above. Thus, to implement a scientist for languages the arguments of TM are construed as members of SEQ, thought of as codes for N. Now recall that an acceptable programming system  j i was fixed, once and for all, in Chapter 2 (see page 2). The foregoing conventions about SEQ and SEG allow us to interpret the indexes i of this system as also referring to functions with domain SEQ or SEG. In particular, when discussing language identification we often let  j i refer to scientists with domain SEQ. And when discussing function identification, we often let  j i refer to scientists with domain SEG. The first enumeration can be conceived as listing all the computable scientists in the language domain. The second lists all the computable scientists in the function domain. We use M, with or without decorations, to denote a computable scientist whose domain (either SEQ or SEG) is determined by context.

		§4.1.2 Names for Paradigms

		In the chapters to follow, many models of empirical inquiry will be defined and analyzed. For the sake of concision, each paradigm will be associated with a symbolic name that

		abbreviates its main features. Such names can be illustrated with the paradigms of

		Chapter 3.

		4.1 Definition

		(a) The class of all identifiable collections of languages (in the sense of Section 3.4) is denoted by Lang.

		(b) The class of all identifiable collections of functions (in the sense of Section 3.9.4) is denoted by Func.

		Thus, .1 To illustrate the use of this notation, Exercise 3-3 and Corollary 3.28 imply that . For another illustration, Proposition 3.43 implies that Func is the power set of . It will be convenient to use this kind

		1 So Lang is a set of sets of sets of numbers.