Addison [1] and the Revised Analogies

Kleene’s Theorem 5.3.4 is an immediate consequence of the following more general

Theorem 5.3.6 (Strong Separation for S}, Addison [1]) For any two disjoint, S} subsets of N, there is a HYP set C which separates them, i.e.,

In fact, Addison [1] claims more and less than this result: he states it for subsets A, B of any product space Nn x Nm rather than just N and his (abbreviated) proof is formulated quite abstractly and also gives the classical Separation Theorem 5.1.2 for Y 1; but he does not note that the result holds uniformly (in given S{-codes of A and B), which it does, and he only says of the separating set C that “it is A1” skipping the punchline “and hence HYP” which he certainly knows for subsets of N. This may be partly because there was no generally accepted definition of HYP subset of Nn x Nm at the time, or because Addison’s paper is about separation and not construction principles. He also does not discuss the obvious revision of the analogies (5.7)

which are the working hypotheses of Mostowski [39]. They are bolstered by the following result which is not hard to prove using Spector-type ordinal assignments and the method of proof of Kleene’s Theorem 5.1.3:

Theorem 5.3.7 There exists disjoint n-sets A, B which are not HYP-separable, i.e., no HYP set C satisfies

On the other hand, to my knowledge, Addison [1] was first to refer to Effective Descriptive Set Theory, which suggests that more than “analogies” are in play; and he introduced the modern lightface,... and boldface ? 1... notation which has been universally accepted.

 
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