∏ 0 1 Classes and Degrees of Theories

Jockusch, Carl G.; Soare, Robert I.

doi:10.2307/1996261

Cited by 206 publications

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“…As in [5] Proof. This follows since the random reals form a union of Π 0 1 -classes (given by a universal Martin-Löf test).…”

Section: Super-low Realsmentioning

confidence: 94%

“…Then {e} A (e) can become undefined at most e times. Thus, if we let g(e, s) = 1 when {e} A (e) converges at stage s and g(e, s) = 0 otherwise, then g is an approximation as in definition 1.1, where b(e) = e. The low basis theorem of Jockusch and Soare [5] can also be strengthened to "super-low": each non-empty Π 0 1 class has a super-low member (Proposition 3.1 below). Jump-traceable reals.…”

Section: Definition 11 the Real A Is Super-low If A ≤ Tt ∅ Equivalmentioning

confidence: 99%

“…Jockusch and Soare [5] proved that each nonempty Π 0 1 class has a low member. An analisis of their proof yields Proposition 3.1 Each non-empty Π 0 1 class has a super-low member.…”

Section: Super-low Realsmentioning

confidence: 99%

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Reals which compute little

Nies¹

2017

Logic Colloquium '02

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Abstract. We investigate combinatorial lowness properties of sets of natural numbers (reals). The real A is super-low if A ≤tt ∅ , and A is jump-traceable if the values of {e} A (e) can be effectively approximated in a sense to be specified. We investigate those properties, in particular showing that super-lowness and jump-traceability coincide within the r.e. sets but none of the properties implies the other within the ω-r.e. sets. Finally we prove that, for any low r.e. set B, there is is a K-trivial set A ≤T B.

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“…As in [5] Proof. This follows since the random reals form a union of Π 0 1 -classes (given by a universal Martin-Löf test).…”

Section: Super-low Realsmentioning

confidence: 94%

Section: Definition 11 the Real A Is Super-low If A ≤ Tt ∅ Equivalmentioning

confidence: 99%

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Reals which compute little

Nies¹

2017

Logic Colloquium '02

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“…It thus follows that the class of sets of Turing degree ≤ T ∅ form a basis for non-empty Π 0 1 -classes (the Kreisel Basis Theorem). This result was subsequently strengthened by Shoenfield [1960] to show that the class of sets of Turing degree < T ∅ forms a basis for non-empty Π 0 1 -classes (the Shoenfield basis theorem) and again by Jockusch and Soare [1972] to shows that the class of sets A such that deg(A ) ≤ T 0 form a basis for non-empty Π 0 1 -classes (the low basis theorem). These results anticipate the formulation of WKL 0 as a formal system in the sense that they provide natural computability-theoretic characterizations of its ω-models.…”

Section: The Constructive Failure Of König's Lemma and The Basis Theomentioning

confidence: 98%

The Prehistory of the Subsystems of Second-Order Arithmetic

Dean

Walsh

2017

The Review of Symbolic Logic

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This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program of Friedman and Simpson. We look in particular at: (i) the long arc from Poincaré to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to Weak König's Lemma, and (iv) the large-scale intellectual backdrop to arithmetical transfinite recursion in descriptive set theory and its effectivization by Borel, Lusin, Addison, and others.

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“…Proof. By Theorem 3 of [9] (which gives more than we need) there is a consistent complete extension T of Peano arithmetic in which the only semi-representable relations are either recursive or non-arithmetical. Define C as above.…”

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confidence: 99%

Partial Recursive Functions and Finality

Plotkin

2013

Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky

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Abstract. We seek universal categorical conditions ensuring the representability of all partial recursive functions. In the category Pfn of sets and partial functions, the natural numbers provide both an initial algebra and a final coalgebra for the functor 1 + −. We recount how finality yields closure of the partial functions on natural numbers under Kleene's µ-recursion scheme. Noting that Pfn is not cartesian, we then build on work of Paré and Román, obtaining weak initiality and finality conditions on natural numbers algebras in monoidal categories that ensure the (weak) representability of all partial recursive functions. We further obtain some positive results on strong representability. All these results adapt to Kleisli categories of cartesian categories with natural numbers algebras. However, in general, not all partial recursive functions need be strongly representable.

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∏ 0 1 Classes and Degrees of Theories

Cited by 206 publications

References 0 publications

Reals which compute little

Reals which compute little

The Prehistory of the Subsystems of Second-Order Arithmetic

Partial Recursive Functions and Finality

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