We consider the informal concept of "computability" or "effective calculability" and two of the formalisms commonly used to define it, "(Turing) computability" and "(general) recursiveness". We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. After a careful (...) historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of "computability" and "recursion." Specifically we recommend that: the term "recursive" should no longer carry the additional meaning of "computable" or "decidable;" functions defined using Turing machines, register machines, or their variants should be called "computable" rather than "recursive;" we should distinguish the intensional difference between Church's Thesis and Turing's Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be "Computability Theory" or simply Computability rather than "Recursive Function Theory.". (shrink)

The set of minimal indices of a Gödel numbering $\varphi$ is defined as ${\rm MIN}_{\varphi} = \{e: (\forall i < e)[\varphi_i \neq \varphi_e]\}$ . It has been known since 1972 that ${\rm MIN}_{\varphi} \equiv_{\mathrm{T}} \emptyset^{\prime \prime }$ , but beyond this ${\rm MIN}_{\varphi}$ has remained mostly uninvestigated. This paper collects the scarce results on ${\rm MIN}_{\varphi}$ from the literature and adds some new observations including that ${\rm MIN}_{\varphi}$ is autoreducible, but neither regressive nor (1,2)-computable. We also study several variants of (...) ${\rm MIN}_{\varphi}$ that have been defined in the literature like size-minimal indices, shortest descriptions, and minimal indices of decision tables. Some challenging open problems are left for the adventurous reader. (shrink)

It is natural to wish to study miniaturisations of Cohen forcing suitable to sets of low arithmetic complexity. We consider extensions of the work of Schaeffer [9] and Jockusch and Posner [6] by looking at genericity notions within the Δ2 sets. Different equivalent characterisations of 1-genericity suggest different ways in which the definition might be generalised. There are two natural ways of casting the notion of 1-genericity: in terms of sets of strings and in terms of density functions, as we (...) will see here. While these definitions coincide at the first level of the difference hierarchy, they turn out to differ at other levels. Furthermore, these differences remain when the remainder of the Δ02 sets are considered. While the string characterization of 1-genericity collapses at the second level of the difference hierarchy to 2-genericity, the density function definition gives a very interesting hierarchy at level w and above. Both of these results point towards the deep similarities exhibited by the n-c.e. degrees for n ≥ 2. (shrink)