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		We will construct We in stages s = -1, 0, 1, 2, . . .. By the recursion theorem we allow the construction to have access to the desired index e. The construction of We may complete all the stages, or else reach only finitely many of them. In each stage s that is entered we define a number , and a text Ts, and also indicate what new elements are enumerated into We. It will be the case that at the end of every completed, odd-numbered stage s, Ts is a text for N, and for every completed, even-numbered stage s, Ts is a text for the finite part of We enumerated by the end of stage s.

		Stage -1: Set m-1 = 0 and T-1 = 0,1,2, . . .. Leave We empty.

		Stage 2n, for : Suppose that m2n-1 and text T2n-1 for N have been defined. Search for m2n > m2n-1 such that

		If the search for m2n fails to terminate, then the enumeration of We ceases at this stage. In this case, We is finite, and fails to identify the text T2n-1 for N, because for each m > m2n-1,  G  (h(e), T2n-1[m]) is an index for a language that does not properly include content(T2n-1[m]). If the search for m2n is successful, then enumerate the contents of T2n-1[m2n] into We and set text

		Note that T2n is a text for the finite set of numbers enumerated into We up to this stage.

		Stage 2n+1, for : Suppose that text T2n and m2n have been defined. Search for m2n+1 > m2n such that

		.

		If the search for m2n+1 fails to terminate, then the enumeration of We ceases at this stage. In this case, T2n is a text for the finite language We, and fails to identify T2n because it converges on T2n to all index for a superset of content(T2n) = We. If the search for m2n+1 is successful, then enumerate into We and set text

		Note that T2n+1 is a text for N.

		If the enumeration of We fails to progress past some even stage of the foregoing construction, then We is finite and fails to identify some text for N.