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Computability and recursion
 BULL. SYMBOLIC LOGIC
, 1996
"... We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they b ..."
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We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of “computability” and “recursion.” Specifically we recommend that: the term “recursive ” should no longer carry the additional meaning of “computable” or “decidable;” functions defined using Turing machines, register machines, or their variants should be called “computable” rather than “recursive;” we should distinguish the intensional difference between Church’s Thesis and Turing’s Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be “Computability Theory” or simply Computability rather than
Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a pla ..."
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Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
The history and concept of computability
 in Handbook of Computability Theory
, 1999
"... We consider the informal concept of a “computable ” or “effectively calculable” function on natural numbers and two of the formalisms used to define it, computability” and “(general) recursiveness. ” We consider their origin, exact technical definition, concepts, history, how they became fixed in th ..."
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Cited by 5 (1 self)
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We consider the informal concept of a “computable ” or “effectively calculable” function on natural numbers and two of the formalisms used to define it, computability” and “(general) recursiveness. ” We consider their origin, exact technical definition, concepts, history, how they became fixed in their present roles, and how
Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
, 2009
"... ..."
Arithmetical Sacks Forcing
 Archive for Mathematical Logic
"... Abstract. We answer a question of Jockusch by constructing a hyperimmunefree minimal degree below a 1generic one. To do this we introduce a new forcing notion called arithmetical Sacks forcing. Some other applications are presented. 1. introduction Two fundamental construction techniques in set the ..."
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Abstract. We answer a question of Jockusch by constructing a hyperimmunefree minimal degree below a 1generic one. To do this we introduce a new forcing notion called arithmetical Sacks forcing. Some other applications are presented. 1. introduction Two fundamental construction techniques in set theory and computability theory are forcing with finite strings as conditions resulting in various forms of Cohen genericity, and forcing with perfect trees, resulting in various forms of minimality. Whilst these constructions are clearly incompatible, this paper was motivated by the general question of “How can minimality and (Cohen) genericity interact?”. Jockusch [5] showed that for n ≥ 2, no ngeneric degree can bound a minimal degree, and Haught [4] extended earlier work of Chong and Jockusch to show that that every nonzero Turing degree below a 1generic degree below 0 ′ was itself 1generic. Thus, it seemed that these forcing notions were so incompatible that perhaps no minimal degree could even be comparable with a 1generic one. However, this conjecture was shown to fail independently by Chong and Downey [1] and by Kumabe [7]. In each of those papers, a minimal degree below m < 0 ′ and a 1generic a < 0 ′ ′ are constructed with m < a. The specific question motivating the present paper is one of Jockusch who asked whether a hyperimmunefree (minimal) degree could be below a 1generic one. The point here is that the construction of a hyperimmunefree degree by and large directly uses forcing with perfect trees, and is a much more “pure ” form of SpectorSacks forcing [10] and [9]. This means that it is not usually possible to use tricks such as full approximation or forcing with partial computable trees, which are available to us when we only wish to construct (for instance) minimal degrees. For instance, minimal degrees can be below computably enumerable ones, whereas no degree below 0 ′ can be hyperimmunefree. Moreover, the results of Jockusch [5], in fact prove that for n ≥ 2, if 0 < a ≤ b and b is ngeneric, then a bounds a ngeneric degrees and, in particular, certainly is not hyperimmune free. This contrasts quite strongly with the main result below. In this paper we will answer Jockusch’s question, proving the following result.
TURING DEGREES OF REALS OF POSITIVE EFFECTIVE PACKING DIMENSION
"... Abstract. A relatively longstanding question in algorithmic randomness is Jan Reimann’s question whether there is a Turing cone of broken dimension. That is, is there a real A such that {B: B ≤T A} contains no 1random real, yet contains elements of nonzero effective Hausdorff Dimension? We show tha ..."
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Abstract. A relatively longstanding question in algorithmic randomness is Jan Reimann’s question whether there is a Turing cone of broken dimension. That is, is there a real A such that {B: B ≤T A} contains no 1random real, yet contains elements of nonzero effective Hausdorff Dimension? We show that the answer is affirmative if Hausdorff dimension is replaced by its inner analogue packing dimension. We construct a minimal degree of effective packing dimension 1. This leads us to examine the Turing degrees of reals with positive effective packing dimension. Unlike effective Hausdorff dimension, this is a notion of complexity which is shared by both random and sufficiently generic reals. We provide a characterization of the c.e. array noncomputable degrees in terms of effective packing dimension. 1.
Computably Enumerable Vector Spaces, Dependence Relations, and Turing Degrees By
, 2002
"... I would like to thank my advisor, Valentina Harizanov, for all her support and help. She worked with me and encouraged my research and studies throughout the Ph.D. program. She has been my mentor, collaborator, and a good friend. IamgratefultoRodDowney,AliEnayat,AndreiMorozov,MichaelMoses, and Frank ..."
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I would like to thank my advisor, Valentina Harizanov, for all her support and help. She worked with me and encouraged my research and studies throughout the Ph.D. program. She has been my mentor, collaborator, and a good friend. IamgratefultoRodDowney,AliEnayat,AndreiMorozov,MichaelMoses, and Frank Stephan for the useful mathematical discussions. I am also indebted to the other committee members, Michele Friend and Irving Glick, for their suggestions on the final form of my thesis. I would like to thank the Faculty members and graduate students from the Department of Mathematics for their various support and for organizing various stimulating seminars and discussions. I would like to thank my wife Anna for supporting my family and being patient during my long days and nights these years. We present some structural theorems on the lattice L(V∞) of computably enumerable vector spaces and its factorlattices. The reader who is not familiar with
Gödel's Incompleteness Theorems: A Revolutionary View of the Nature of Mathematical Pursuits
"... The work of the mathematician Kurt Gödel changed the face of mathematics forever. His famous incompleteness theorem proved that any formalized system of mathematics would always contain statements that were undecidable, showing that there are certain inherent limitations to the way many mathematicia ..."
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The work of the mathematician Kurt Gödel changed the face of mathematics forever. His famous incompleteness theorem proved that any formalized system of mathematics would always contain statements that were undecidable, showing that there are certain inherent limitations to the way many mathematicians studies mathematics. This paper provides a history of the mathematical developments that laid the foundation for Gödel's work, describes the unique method used by Gödel to prove his famous incompleteness theorem, and discusses the farreaching mathematical implications thereof. 2 I.
LOW LEVEL NONDEFINABILITY RESULTS: DOMINATION AND RECURSIVE ENUMERATION
"... Abstract. We study low level nondefinability in the Turing degrees. We prove a variety of results, including for example, that being array nonrecursive is not definable by a Σ1 or Π1 formula in the language (≤,REA) where REA stands for the “r.e. in and above ” predicate. In contrast, this property i ..."
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Abstract. We study low level nondefinability in the Turing degrees. We prove a variety of results, including for example, that being array nonrecursive is not definable by a Σ1 or Π1 formula in the language (≤,REA) where REA stands for the “r.e. in and above ” predicate. In contrast, this property is definable by a Π2 formula in this language. We also show that the Σ1theory of (D, ≤,REA) is decidable. 1.
Computability Theory, Algorithmic Randomness and Turing’s Anticipation
"... Abstract. This article looks at the applications of Turing’s Legacy in computation, particularly to the theory of algorithmic randomness, where classical mathematical concepts such as measure could be made computational. It also traces Turing’s anticipation of this theory in an early manuscript. 1 ..."
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Abstract. This article looks at the applications of Turing’s Legacy in computation, particularly to the theory of algorithmic randomness, where classical mathematical concepts such as measure could be made computational. It also traces Turing’s anticipation of this theory in an early manuscript. 1