The strong weak truth table (sw) reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, alternative to the Solovay reducibility. It also occurs naturally in proofs in classical computability theory as well as in the recent work of Soare, Nabutovsky, and Weinberger on applications of computability to differential geometry. We study the sw-degrees of c.e. reals and construct a c.e. real which has no random c.e. real (i.e., Ω number) sw-above it.
We show that for any computably enumerable set A and any equation image set L, if L is low and equation image, then there is a c.e. splitting equation image such that equation image. In Particular, if L is low and n-c.e., then equation image is n-c.e. and hence there is no low maximal n-c.e. degree.
Khutoretskii's Theorem states that the Rogers semilattice of any family of c.e. sets has either at most one or infinitely many elements. A lemma in the inductive step of the proof shows that no Rogers semilattice can be partitioned into a principal ideal and a principal filter. We show that such a partitioning is possible for some family of d.c.e. sets. In fact, we construct a family of c.e. sets which, when viewed as a family of d.c.e. sets, has (up (...) to equivalence) exactly two computable Friedberg numberings ¼ and ν, and ¼ reduces to any computable numbering not equivalent to ν. The question of whether the full statement of Khutoretskii's Theorem fails for families of d.c.e. sets remains open. (shrink)
We show that in the c.e. weak truth table degrees if b < c then there is an a which contains no hypersimple set and b < a < c. We also show that for every w < c in the c.e. wtt degrees such that w is hypersimple, there is a hypersimple a such that w < a < c. On the other hand, we know that there are intervals which contain no hypersimple set.
We show that the Q-degree of a hyperhypersimple set includes an infinite collection of Q 1-degrees linearly ordered under ${\leq_{Q_1}}$ with order type of the integers and consisting entirely of hyperhypersimple sets. Also, we prove that the c.e. Q 1-degrees are not an upper semilattice. The main result of this paper is that the Q 1-degree of a hemimaximal set contains only one c.e. 1-degree. Analogous results are valid for ${\Pi_1^0}$ s 1-degrees.
It is shown that there exists a low2 Harrington non-splitting base-that is, a low2 computably enumerable (c.e.) degree a such that for any c.e. degrees x, y, if $0' = x \vee y$ , then either $0' = x \vee a$ or $0' = y \vee a$ . Contrary to prior expectations, the standard Harrington non-splitting construction is incompatible with the $low_{2}-ness$ requirements to be satisfied, and the proof given involves new techniques with potentially wider application.
Let [NB]₁ denote the ideal generated by nonbounding c.e. degrees and NCup the ideal of noncuppable c.e. degrees. We show that both [NB]₁ ∪ NCup and the ideal generated by nonbounding and noncuppable degrees are new, in the sense that they are different from M, [NB]₁ and NCup—the only three known definable ideals so far.
We show that for every c.e. degree a > 0 there exists an intrinsically c.e. relation on the domain of a computable structure whose degree spectrum is {0, a}. This result can be extended in two directions. First we show that for every uniformly c.e. collection of sets S there exists an intrinsically c.e. relation on the domain of a computable structure whose degree spectrum is the set of degrees of elements of S. Then we show that if α ∈ (...) ω∪{ω } then for any α-c.e. degree a > 0 there exists an intrinsically α-c.e. relation on the domain of a computable structure whose degree spectrum is {0, a}. All of these results also hold for m-degree spectra of relations. (shrink)
Computably Lipschitz reducibility , was suggested as a measure of relative randomness. We say α≤clβ if α is Turing reducible to β with oracle use on x bounded by x+c. In this paper, we prove that for any non-computable real, there exists a c.e. real so that no c.e. real can cl-compute both of them. So every non-computable c.e. real is the half of a cl-maximal pair of c.e. reals.
In this paper, we will prove that the plus cupping degrees generate a definable ideal on c.e. degrees different from other ones known so far, thus answering a question asked by Li and Yang (Proceedings of the 7th and the 8th Asian Logic Conferences. World Scientific Press, Singapore, 2003).
We prove that there is a ${\mathrm{\Delta }}_{2}^{0}$ , 1-random set Y such that every computably enumerable set which is computable from Y is strongly jump-traceable. We also show that for every order function h there is an ω-c.e. random set Y such that every computably enumerable set which is computable from Y is h-jump-traceable. This establishes a correspondence between rates of jump-traceability and computability from ω-c.e. random sets.
In this paper, we study the bi-isolation phenomena in the d.c.e. degrees and prove that there are c.e. degrees c₁ < c₂ and a d.c.e. degree d ∈ (c₁, c₂) such that (c₁, d) and (d, c₂) contain no c.e. degrees. Thus, the c.e. degrees between c₁ and c₂ are all incomparable with d. We also show that there are d.c.e. degrees d₁ < d₂ such that (d₁, d₂) contains a unique c.e. degree.
In this paper we study structural properties of n-c. e. Q-degrees. Two theorems contain results on the distribution of incomparable Q-degrees. In another theorem we prove that every incomplete Q-degree forms a minimal pair in the c. e. degrees with a Q-degree. In a further theorem it is proved that there exists a c. e. Q-degree that is not half of a minimal pair in the c. e. Q-degrees.
The detrimental effect of the public recital on the quality of epic production in the first century is a stock theme both in ancient and in modern literary criticism. While previous studies on the epic recital emphasize its negative effects, or aim at its reconstruction as social reality, I focus on its conflicting representations by the ancients themselves and the lessons that we can learn from them. The voices of critics and defenders reveal anxieties about who controls the prestigious high (...) genre of epic and about the construction of gender and social status through epic's public performance. Critics of the recitatio such as Horace, Persius, Petronius and Juvenal represent it as an informal and popular event that panders to public taste and incurs infamy. These critics charge that the epic recital has an effeminizing effect both on the recitator and on his audience, a charge traditionally advanced against orators and actors. Because the epic recital in Rome lacks a performative context similar to the rhapsodic performances in Greece, its public image mirrors that of other public performances . The main point of divergence is the presence of the book, the symbol of permanent fame, which casts upon the recital the shadow of an evanescent entertainment. Defenders see the recital as instrumental to the maintenance of literature's value in society. This means that well into the first century there were those who felt nostalgia for the epic recital as a bulwark of male aristocratic values and who wanted to reclaim its prestige. Both Statius as a poet, who performed regularly at Domitian's court, and Suetonius, who acted as Hadrian's ab epistulis some twenty years after Statius' death, defend the waning reputation of the contemporary epic recital in an effort to reclaim it as a prestigious component of imperial literary culture. While critics unanimously negate the recital's role in the achievement of poetic fame in favor of the book, Suetonius chooses anecdotal evidence about the early performances by grammarians grammarians that show the important role of the recital in forging poetic fame. The emperor and the members of the new aristocracy whom Statius explicitly names as the target audience of his epic recitals have a stake in reclaiming for imperial culture an institution that went back to the second century B.C.E. and carried the prestige of an aristocratic event. (shrink)
A real α is called a c. e. real if it is the halting probability of a prefix free Turing machine. Equivalently, α is c. e. if it is left computable in the sense that L = {q ∈ ℚ : q ≤ α} is a computably enumerable set. The natural field formed by the c. e. reals turns out to be the field formed by the collection of the d. c. e. reals, which are of the form α—β, where (...) α and β are c. e. reals. While c. e. reals can only be found in the c. e. degrees, Zheng has proven that there are Δ02 degrees that are not even n-c. e. for any n and yet contain d. c. e. reals, where a degree is n-c. e. if it contains an n-c. e. set. In this paper we will prove that every ω-c. e. degree contains a d. c. e. real, but there are ω + 1-c. e. degrees and, hence Δ02 degrees, containing no d. c. e. real. (shrink)
Cooper proved in [S.B. Cooper, Strong minimal covers for recursively enumerable degrees, Math. Logic Quart. 42 191–196] the existence of a c.e. degree with a strong minimal cover . So is the greastest c.e. degree below . Cooper and Yi pointed out in [S.B. Cooper, X. Yi, Isolated d.r.e. degrees, University of Leeds, Dept. of Pure Math., 1995. Preprint] that this strongly minimal cover cannot be d.c.e., and meanwhile, they proposed the notion of isolated degrees: a d.c.e. degree is isolated (...) by a c.e. degree if is the greatest c.e. degree below , and we also say that isolates . In [G. Wu, Bi-isolation in the d.c.e. degrees, J. Symbolic Logic 69 409–420], Wu extended Cooper–Yi’s notion and proved that there are intervals of d.c.e. degrees containing exactly one c.e. degree . Following Cooper and Yi’s notion, is called a bi-isolating degree. The bi-isolating degrees are dense in the high c.e. degrees. Arslanov asked whether the bi-isolating degrees occur in every jump class. In this paper, we prove that there are low bi-isolating degrees, providing a partial solution to Arslanov’s question. (shrink)
In 1926, C. E. Ayres, a young assistant editor of The New Republic, had completed a draft of his first book, Science: The False Messiah. His publishers, Bobbs-Merrill, were enthusiastic but also somewhat worried—the book, which was a blistering critique of the public understanding of science, was engagingly written and eminently readable, but it was also provocative. Bobbs-Merrill were concerned that Ayres’ “very saucy” approach might damage sales, especially given that he was a complete unknown as far as the general (...) public was concerned; and in order to boost Ayres’ credibility and, hence, future sales of the book, they felt that he needed an endorsement.1 Ayres thus dutifully set about writing to his friendly .. (shrink)
The notion of superhigh computably enumerable (c.e.) degrees was first introduced by (Mohrherr in Z Math Logik Grundlag Math 32: 5–12, 1986) where she proved the existence of incomplete superhigh c.e. degrees, and high, but not superhigh, c.e. degrees. Recent research shows that the notion of superhighness is closely related to algorithmic randomness and effective measure theory. Jockusch and Mohrherr proved in (Proc Amer Math Soc 94:123–128, 1985) that the diamond lattice can be embedded into the c.e. tt-degrees preserving 0 (...) and 1 and that the two atoms can be low. In this paper, we prove that the two atoms in such embeddings can also be superhigh. (shrink)
Call a set A n-correctable if every set Turing reducible to A via a Turing machine that on any input makes at most n queries is Turing reducible to A via a Turing machine that on any input makes at most n-queries and on any input halts no matter what answers are given to its queries. We show that if a c.e. set A is n-correctable for some n ≥ 2, then it is n-correctable for all n. We show that (...) this is the optimal such result by constructing a c.e. set that is 1-correctable but not 2-correctable. The former result is obtained by examining the logical closure properties of c.e. sets that are 2-correctable. (shrink)
It is proved that the (2p)-c. e. e-degrees are not elementarily equivalent to the (2p + 1)-c. e. e-degrees for each nonzero p ∈ ω. It follows that m-c. e. e-degrees are not elementarily equivalent to the n-c. e. e-degrees if 1 < m < n.
A completely mitotic computably enumerable degree is a c.e. degree in which every c.e. set is mitotic, or equivalently in which every c.e. set is autoreducible. There are known to be low, low2, and high completely mitotic degrees, though the degrees containing non-mitotic sets are dense in the c.e. degrees. We show that there exists an upper cone of c.e. degrees each of which contains a non-mitotic set, and that the completely mitotic c.e. degrees are nowhere dense in the c.e. (...) degrees. We also show that there is a set computably enumerable in and above 0′ which is not the jump of any completely mitotic degree. (shrink)
We construct a high c.e. degree which is not the join of two minimal degrees and so refute Posner's conjecture that every high c.e. degree is the join of two minimal degrees. Additionally, the proof shows that there is a high c.e. degree a such that for any splitting of a into degrees b and c one of these degrees bounds a 1-generic degree.
E. Husserl, Logik und allgemeine Wissenschaftstheorie. Vorlesungen 1917/18, mit ergänzenden Texten aus der ersten Fassung 1910/11. Introduction by U. Panzer. Dordrecht:Kluwer, 1996. lxii + 554 pp. £130. ISBN0 792 33731 X D. Jacquette, Meinongian logic. The semantics of existence and nonexistence. Berlin and New York:Walter de Gruyter, 1996. xiii + 297 pp. DM 198. ISBN 3 11 014865 X M. Beaney, The Frege Reader. Blackwell Publishers, 1997. xv + 409 pp. £14.99/$21.95. ISBN 0 631 194 452 Elliott Mendelson, Introduction to (...) formal logic, fourth edition. London:Chapman & Hall, 1997. x + 440 pp. £45.00. ISBN 1 412 808307 Samuel Guttenplan, The languages of logic. An introduction to formal logic, second edition. Oxford:Blackwell, 1997. x + 429 pp. No price stated. ISBN 1 55786 988 X A.C. Grayling, An introduction to philosophical logic, third edition. Oxford:Blackwell, 1997. vii + 343 pp. £15.99/$27.99. ISBN 0 631 19982 9 Lewis Carroll, Das Spiel der Logik. Edited with an afterword by P. Good. Translated by M.Zöllner. Cologne:Propen Verlag/frommann-holzboog, 1997. 120pp. 32 DM. ISBN 3 7728 1998 2. (shrink)
In Alabaster v. Barclays Bank plc and Secretary of State for Social Security (No. 2: [2005] E.W.C.A Civ. 508, [2005] I.R.L.R. 576.) Michelle Alabaster won a grand total of £204.53 (plus £65.86 interest) after eight years of litigation, which included two visits to the Court of Appeal and one to the European Court of Justice. This marathon resulted from the sex discrimination which Alabaster had alleged in relation to the calculation of her Statutory Maternity Pay (S.M.P.) whilst she was pregnant (...) 10 years earlier. The technicalities of the statutory schemes involved should not be allowed to disguise the important principle which finally emerges in the Court of Appeal and which underlines one of the longstanding criticisms of the equality legislation, namely the requirement that a woman must compare herself with a man in order to establish unlawful sex discrimination. (shrink)
