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		countably infinite. denotes the concatenation of  s  and . We also use to denote the addition of the pair (x, y) at the end of  s . We introduce one special piece of notation for SEG; we let be the finite function from N to N whose graph is content( s ). Thus provided (x, y) occurs in  s .

		As in the language context, a scientist examining a text G may be conceived as entering "evidential state" G[n] at moment n. So it is natural to take a scientist to be any system that maps the class of all possible evidential states — namely, SEG — into the set of all hypotheses, namely, N. Officially, a scientist within the function identification paradigm is any mapping — partial or total, computable or noncomputable — from SEG to N. As before, we use F as a variable for scientists, allowing context to determine whether F stands for scientists over SEQ or SEG.

		Let text G and be such that . Then denotes the function corresponding to the program that F emits upon examining the finite sequence of length n in G. We often say that this function is "conjectured by F on G[n]." If , then is not defined. Note that need not be total.

		§3.9.4 Function Identification: Scientific Success

		The criterion of success associated with our second paradigm is a straightforward adaptation of the criterion for language identification given in Section 3.4 above.

		3.42 Definition (Gold [80]) Let scientist F, text G, , and be given.

		(a) F converges on G to (written: ) just in case for all but finitely many , F(G[n]) = i. If there exists an i such that , then we say that ; otherwise we say that F(G) diverges (written: ).

		(b) F identifies G just in case there is such that F converges to j on G, and .

		(c) F identifies f just in case F identifies every text for f.

		(d) F identifies C just in case F identifies every , C is said to be identifiable just in case some scientist identifies it; it is said to be unidentifiable otherwise.

		§3.10 Characterization of Identifiable

		Identifiability in the current paradigm turns out to be a trivial matter, inasmuch as every subset of is identifiable. This is the content of the next theorem.