Results 1 
7 of
7
An Overview of the Computably Enumerable Sets
"... The purpose of this article is to summarize some of the results on the algebraic structure of the computably enumerable (c.e.) sets since 1987 when the subject was covered in Soare 1987 , particularly Chapters X, XI, and XV. We study the c.e. sets as a partial ordering under inclusion, (E; `). We do ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
The purpose of this article is to summarize some of the results on the algebraic structure of the computably enumerable (c.e.) sets since 1987 when the subject was covered in Soare 1987 , particularly Chapters X, XI, and XV. We study the c.e. sets as a partial ordering under inclusion, (E; `). We do not study the partial ordering of the c.e. degrees under Turing reducibility, although a number of the results here relate the algebraic structure of a c.e. set A to its (Turing) degree in the sense of the information content of A. We consider here various properties of E: (1) deønable properties; (2) automorphisms; (3) invariant properties; (4) decidability and undecidability results; miscellaneous results. This is not intended to be a comprehensive survey of all results in the subject since 1987, but we give a number of references in the bibliography to other results.
Definable properties of the computably enumerable sets
 Proceedings of the Oberwolfach Conference on Computability Theory
, 1996
"... Post 1944 began studying properties of a computably enumerable (c.e.) set A such as simple, hsimple, and hhsimple, with the intent of finding a property guaranteeing incompleteness of A. From observations of Post 1943 and Myhill 1956, attention focused by the 1950's on properties definable in the ..."
Abstract
 Add to MetaCart
Post 1944 began studying properties of a computably enumerable (c.e.) set A such as simple, hsimple, and hhsimple, with the intent of finding a property guaranteeing incompleteness of A. From observations of Post 1943 and Myhill 1956, attention focused by the 1950's on properties definable in the inclusion ordering of c.e. subsets of!, namely E = (fWngn2! ; ae). In the 1950's and 1960's Tennenbaum, Martin, Yates, Sacks, Lachlan, Shoenfield and others produced a number of elegant results relating Edefinable properties of A, like maximal, hhsimple, atomless, to the information content (usually the
Extensions, Automorphisms, and Definability
 CONTEMPORARY MATHEMATICS
"... This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to ..."
Abstract
 Add to MetaCart
This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to know the dynamical properties of a c.e. set We , not merely whether an element x is enumerated in We but when, relative to its appearance in other c.e. sets. We present here
Definability and Automorphisms of the Computably Enumerable Sets
, 2010
"... The computably enumerable (c.e.) sets have been central to computability theory since its inception. We study the structure of the c.e. sets, which forms a lattice E under set inclusion. Jump classes, such as the low degrees, allow us to classify the c.e. sets according to their information content. ..."
Abstract
 Add to MetaCart
The computably enumerable (c.e.) sets have been central to computability theory since its inception. We study the structure of the c.e. sets, which forms a lattice E under set inclusion. Jump classes, such as the low degrees, allow us to classify the c.e. sets according to their information content. The upward closed jump classes Ln and Hn have all been shown to be definable by a latticetheoretic formula, except for L1, the nonlow degrees, which is the only jump class whose definability was unknown. We say a class of c.e. degrees is invariant if it is the set of degrees of a class of c.e. sets that is invariant under automorphisms of E. All definable classes of degrees are invariant. We show that L1 is in fact noninvariant, thus proving a 1996 conjecture of Harrington and Soare in [3] that the nonlow degrees are not definable, and completing the problem of determining the definability of each jump class. 1
THE INFINITE INJURY PRIORITY METHOD1
"... JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JS ..."
Abstract
 Add to MetaCart
JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.
Review of Peter Cholak, “Automorphisms of the Lattice of Recursively Enumerable Sets ” Memoirs of the American Math. Soc. (1995) viii+151 pp and
"... Historical Origins. Computability (or recursion) theory grew from our efforts to understand the algorithmic content of mathematics. One of the great achievements of the 20th century is the development of a precise formulation of the notion of a computable function via the ChurchTuring Thesis. In an ..."
Abstract
 Add to MetaCart
Historical Origins. Computability (or recursion) theory grew from our efforts to understand the algorithmic content of mathematics. One of the great achievements of the 20th century is the development of a precise formulation of the notion of a computable function via the ChurchTuring Thesis. In an very influential paper [Po44], Post articulated some of the fundamental notions at the heart of most undecidability proofs. He observed that these proofs worked by coding some “noncomputability ” into the theory at hand thereby arguing that the relevant structures could emulate computation. One key concept was that of effective enumeration which leads to the notion of a computably (recursively) enumerable set. A computably enumerable set is a subset of N which is the range of a computable (total) function. The intuitive idea is that if f is computable, then I can “effectively list ” (not necessarily in order) the range of f as {f(0), f(1),...}. Think of consequences of a computably enumerable set of axioms for a formal system. The other key concept discovered by Turing [1939] and further developed in Post’s paper