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Effective Mathematics of the Uncountable by Joel David Hamkins, Denis Hirschfeldt, Noam Greenberg

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E-RECURSIVE INTUITIONS

GERALD E. SACKS

In MemoriamProfessor Joseph R. Shoenfield

Abstract An informal sketch (with intermittent details) of parts of E-Recursion theory, mostly old, some new, that stresses intuition. The lack of effective unbounded search is balanced by the availability of divergence witnesses. A set is E-closed iff it is transitive and closed under the application of partial E-recursive functions. Some finite injury, forcing, and model theoretic constructions can be adapted to E-closed sets that are not Σ1 admissible. Reflection plays a central role.

§1. Initial Intuitions. One of the central intuitions of classical recursion theory is the effectiveness of unbounded search. Let A be a nonempty recursively enumerable set of nonnegative ...

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