On a Conjecture of Kleene and Post

Abstract

A proof is given that 0 (the largest Turing degree containing a computably enumerable set) is definable in the structure of the degrees of unsolvability. This answers a long-standing question of Kleene and Post, and has a number of corollaries including the definability of the jump operator.Mathematics Subject Classification: 03D25, 03D30, 03D35.Following Gödel [8], Church [1] and Turing's [33] discovery that most familiar mathematical theories are undecidable, the existence of a noncomputable universe intimately connected with the relatively small world of everyday mathematics has led to the development (see Kleene and Post [14]) of the degrees of unsolvability D as an appropriate theoretical framework, or fine structure theory. While more mundane considerations dictated a direct interest in the details of the computable universe, the early work of Kleene and Post [14] presaged an ever deepening understanding of the nature of the noncomputable universe, a development both of philosophical significance and of wider technical application.The most important noncomputable degree 0 is that containing the (coded) undecidable axiomatic theories of Gödel, as well as many other natural mathematical objects. When relativised to a set A of degree a, it gives rise to a jump operator taking a to a strictly higher degree a , defined as the largest degree containing sets which are computably enumerable (written c. e.) using oracle A. Post's Theorem [23] showed a close relationship between the quantifier forms of most naturally occuring sets of numbers and the ascending sequence 0 < 0 < 0 < · · · < 0 (n+1) = (0 (n) ) < · · · (0 being the degree of the computable sets). 0 and the jump operator are basic

A splitting theorem for $n-REA$ degrees

“…set is of 2−REA degree (see Jockusch and Shore [1984] for a proof of a more general fact), it suffices to prove that every 2 − REA degree d is splittable over every a avoiding any b to show that there is no d − r.e. set as claimed in the Main Theorem of Cooper [1990], [1993]. In fact, we prove this for every n − REA degree d. The proof is not uniform in the sense that given D, A and B we do not construct a splitting (D 0 , D 1 ) of D that lies above A and not above B in such a way as to effectively produce indices for the D i from those of the given sets.…”

Degrees of Unsolvability

“…In order to define 0 ′ , which by relativization would give the definition of the jump, Cooper considers some splitting properties and he introduces the notion of a degree d being relatively splittable over a predecessor a. In [Cooper, 2001], Cooper attempts an amended proof of the definability of the jump using a variant of his proposed definition of 1990. By relativizing the Sacks Splitting Theorem, 0 ′ has this property.…”

“…He claims that 0 ′ is the greatest degree x with the property that, for all a, x∨a is relatively splittable over a. (See for instance Jockusch's review [Jockusch, 2002] of [Cooper, 2001] and the discussion in [Shore, 2006].) The proposed proof of maximality has two parts.…”

Beyond Gödel's Theorem: Turing Nonrigidity Revisited

“…For instance, it is Turing definability rather than the more familiar notions of provability, and completeness of axiomatic theories, which are more relevant to an analysis of the scope of scientific understanding in the real world; while recent results concerning Turing invariance and nonrigidity have both negative and positive consequences for science as a means to knowledge. Turing nonrigidity (see [6]) may reinforce scepticism about a narrow perspective based on scientific observation, as modelled by Turing computable processes: but the proliferation of invariant substructures of the Turing universe (see, for example, Cooper [5], Nies, Shore and Slaman [27] or Odifreddi [28]) can be viewed as reflecting negatively on the more radical postmodernist and (post-) structuralist views of the roles of culture and language in relation to science (cf. Gross and Levitt [17]) -objective reality does exist.…”

“…The form of the noninvariance attached to subatomic individual states suggested by the Heisenberg [21] Uncertainty Principle (but not necessarily to non-unary relations, such as those derived from the weak and the strong nuclear forces) would lead one to expect many local relations, but not singletons, rigid relative to the global model. The former is borne out by Cooper [5], Jockusch and Shore [23], Nerode and Shore [26] and Nies, Shore and Slaman [27], for instance, providing a rich source of subatomic structure, but there appear to be no invariant computably enumerable singletons other than 0 and 0 ′ . It is worth noting that the relationship between invariance and definability is not well understood, so it may be that there are well-defined elements of the quantum environment which cannot be theoretically captured in a framework derived (inductively or otherwise) from scientific observation.…”