We study arithmetic and hyperarithmetic degrees of categoricity. We extend a result of E. Fokina, I. Kalimullin, and R. Miller to show that for every computable ordinal $\alpha$, $\mathbf{0}^{}$ is the degree of categoricity of some computable structure $\mathcal{A}$. We show additionally that for $\alpha$ a computable successor ordinal, every degree $2$-c.e. in and above $\mathbf{0}^{}$ is a degree of categoricity. We further prove that every degree of categoricity is hyperarithmetic and show that the index set of structures with degrees (...) of categoricity is $\Pi_{1}^{1}$-complete. (shrink)

We give several characterizations of Schnorr trivial sets, including a new lowness notion for Schnorr triviality based on truth-table reducibility. These characterizations allow us to see not only that some natural classes of sets, including maximal sets, are composed entirely of Schnorr trivials, but also that the Schnorr trivial sets form an ideal in the truth-table degrees but not the weak truth-table degrees. This answers a question of Downey, Griffiths and LaForte.

We show that every countable ideal of degrees that are low for isomorphism is contained in a principal ideal of degrees that are low for isomorphism by adapting an exact pair construction. We further show that within the hyperimmune free degrees, lowness for isomorphism is entirely independent of computable traceability.

We study the connection between Schnorr triviality and genericity. We show that while no 2-generic is Turing equivalent to a Schnorr trivial and no 1-generic is tt-equivalent to a Schnorr trivial, there is a 1-generic that is Turing equivalent to a Schnorr trivial. However, every such 1-generic must be high. As a corollary, we prove that not all K-trivials are Schnorr trivial. We also use these techniques to extend a previous result and show that the bases of cones of Schnorr (...) trivial Turing degrees are precisely those whose jumps are at least 0". (shrink)

We show that van Lambalgen's Theorem fails with respect to recursive randomness and Schnorr randomness for some real in every high degree and provide a full characterization of the Turing degrees for which van Lambalgen's Theorem can fail with respect to Kurtz randomness. However, we also show that there is a recursively random real that is not Martin-Löf random for which van Lambalgen's Theorem holds with respect to recursive randomness.

We investigate the relationship between lowness for Schnorr randomness and Schnorr triviality. We show that a real is low for Schnorr randomness if and only if it is Schnorr trivial and hyperimmune free.

This article is an autoethnographic account of my journey from theatre stage manager to academic stage manager. Performing arts education and training in Higher Education is a diverse field, rangin...

The weakly random reals contain not only the Schnorr random reals as a subclass but also the weakly 1-generic reals and therefore the n -generic reals for every n . While the class of Schnorr random reals does not overlap with any of these classes of generic reals, their degrees may. In this paper, we describe the extent to which this is possible for the Turing, weak truth-table, and truth-table degrees and then extend our analysis to the Schnorr random and (...) hyperimmune reals. (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)

But Williams had created a field of historical study, where his white counterparts had not. Single-handedly and without the blessing or approval of the academic community, Williams had called attention to the importance of including Afro-Americans in any acceptable and comprehensive history of the nation long before the historians of various groups of European-Americans or Asian-Americans had begun to advocate a similar treatment for their groups. And if Williams did not impress the white professional historians, he gave heart and encouragement (...) to future Afro-American historians. When the History of the Negro Troops appeared in 1887, nineteen-year-old W. E. B. Du Bois was a college senior at Fisk University and editor-in-chief of the student magazine, The Fisk Herald. In the columns of the Herald Du Bois wrote, "At last we have a historian; not merely a Negro historian, but a man who judged by his merits alone has written a splendid narrative. The Herald congratulates George W. Williams, and the race, which may justly be . . . [proud] of him."1 Many years later, Carter G. Woodson, the founder of the Association for the Study of Negro Life and History and of the Journal of Negro History, described Williams' History of the Negro Troops as "one of the most valuable accounts of the Civil War."2 With words like these from Du Bois and Woodson, on whose shoulders much of the second stage of Afro-American historiography would rest, it is not too much to say that George Washington Williams was responsible for the beginnings of Afro-American historiography. · 1. The Fisk Herald, January 1888, p. 8.· 2. Woodson's appraisal of Williams was found among his papers and made available to me by Dr. Charles H. Wesley when he was executive director of the Association for the Study of Negro Life and History, which had been founded by Woodson in 1915. John Hope Franklin, president-elect of the American Historical Society, has written a biography of George Washington Williams. He is the John Matthews Manly Distinguished Service Professor of History at the University of Chicago and the author of, among other works, From Slavery to Freedom: A History of Negro Americans and A Southern Odyssey: Travelers in the Antebellum North. (shrink)

Excerpt from Lincoln and Public Morality Finally, there were the problems related to the prosecution of the war and to the aims for which it was being fought. If this did not bear directly on such matters as graft and dishonesty, it was no less related to the basic problem of public morality. For it was not only desirable but perhaps even necessary for national survival to discuss honestly and forthrightly the war aims. The risks in misrepresentation or dishonesty were (...) formidable, and any leader who was unwilling to enunciate a position on the critical questions of the day was courting personal and national disaster. Lincoln had no intention of doing either. Too long, even before the secession crisis, he had spoken for Union above all, and if he was committee to an ideal of freedom for all it was to be achieved and maintained in an indestructible union. His unequivocal posi tion in this regard was, in a sense, a measure of his high public morality. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

A real is Martin-Löf (Schnorr) random if it does not belong to any effectively presented null ${\Sigma^0_1}$ (recursive) class of reals. Although these randomness notions are very closely related, the set of Turing degrees containing reals that are K-trivial has very different properties from the set of Turing degrees that are Schnorr trivial. Nies proved in (Adv Math 197(1):274–305, 2005) that all K-trivial reals are low. In this paper, we prove that if ${{\bf h'} \geq_T {\bf 0''}}$ , then h (...) contains a Schnorr trivial real. Since this concept appears to separate computational complexity from computational strength, it suggests that Schnorr trivial reals should be considered in a structure other than the Turing degrees. (shrink)

Review by: Johanna N. Y. Franklin The Bulletin of Symbolic Logic, Volume 19, Issue 1, Page 115-118, March 2013.