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The theory of the metarecursively enumerable degrees
"... Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equival ..."
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Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equivalently, that of the truth set of L ω CK
ANALYSIS IN J2
, 2005
"... Abstract. This is an expository paper in which I explain how core mathematics, particularly abstract analysis, can be developed within a concrete countable set J2 (the second set in Jensen’s constructible hierarchy). The implication, wellknown to proof theorists but probably not to most mainstream ..."
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Abstract. This is an expository paper in which I explain how core mathematics, particularly abstract analysis, can be developed within a concrete countable set J2 (the second set in Jensen’s constructible hierarchy). The implication, wellknown to proof theorists but probably not to most mainstream mathematicians, is that ordinary mathematical practice does not require an enigmatic metaphysical universe of sets. I go further and argue that J2 is a superior setting for normal mathematics because it is free of irrelevant settheoretic pathologies and permits stronger formulations of existence results. Perhaps many mathematicians would admit to harboring some feelings of discomfort about the ethereal quality of Cantorian set theory. Yet draconian alternatives such as intuitionism, which holds that simple numbertheoretic statements like the twin primes conjecture may have no definite truth value, probably violate the typical working mathematician’s intuition far more severely than any vague unease he may feel about remote cardinals such as, say, ℵℵω. I believe that ordinary mathematical practice is actually most compatible with
Transfinite Machine Models
, 2011
"... In recent years there has emerged the study of discrete computational models which are allowed to act transfinitely. By ‘discrete ’ we mean that the machine models considered are not analogue machines, but compute by means of distinct stages or in units of time. The paradigm of such models is, of co ..."
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In recent years there has emerged the study of discrete computational models which are allowed to act transfinitely. By ‘discrete ’ we mean that the machine models considered are not analogue machines, but compute by means of distinct stages or in units of time. The paradigm of such models is, of course, Turing’s original
Ordinal Machines and Admissible Recursion Theory
"... We generalize standard Turing machines working in time ω on a tape of length ω to abstract machines with time α and tape length α, for α some limit ordinal. This model of computation determines an associated computability theory: αcomputability theory. We compare the new theory to αrecursion theor ..."
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We generalize standard Turing machines working in time ω on a tape of length ω to abstract machines with time α and tape length α, for α some limit ordinal. This model of computation determines an associated computability theory: αcomputability theory. We compare the new theory to αrecursion theory, which was developed by G. Sacks and his school. For α an admissible ordinal, the basic notions of αcomputable and αrecursive as well as αcomputably enumerable and αrecursively enumerable completely agree. Moreover there is an isomorphism of parts of the degree structure induced by αcomputability and of a degree structure in αrecursion theory, which allows us to transfer, e.g., the SacksSimpson theorem or Shore’s density theorem to αcomputability theory. We emphasize the algorithmic approach by giving a proof of the SacksSimpson theorem, which is solely based on αmachines and does not rely on constructibility theory.