Results 1 
9 of
9
Degrees of random sets
, 1991
"... An explicit recursiontheoretic definition of a random sequence or random set of natural numbers was given by MartinLöf in 1966. Other approaches leading to the notions of nrandomness and weak nrandomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of nrando ..."
Abstract

Cited by 46 (4 self)
 Add to MetaCart
An explicit recursiontheoretic definition of a random sequence or random set of natural numbers was given by MartinLöf in 1966. Other approaches leading to the notions of nrandomness and weak nrandomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of nrandom and weakly nrandom sequences with an emphasis on the structure of their Turing degrees. After an introduction and summary, in Chapter II we present several equivalent definitions of nrandomness and weak nrandomness including a new definition in terms of a forcing relation analogous to the characterization of ngeneric sequences in terms of Cohen forcing. We also prove that, as conjectured by Kurtz, weak nrandomness is indeed strictly weaker than nrandomness. Chapter III is concerned with intrinsic properties of nrandom sequences. The main results are that an (n + 1)random sequence A satisfies the condition A (n) ≡T A⊕0 (n) (strengthening a result due originally to Sacks) and that nrandom sequences satisfy a number of strong independence properties, e.g., if A ⊕ B is nrandom then A is nrandom relative to B. It follows that any countable distributive lattice can be embedded
The Discovery Of My Completeness Proofs
 Bulletin of Symbolic Logic
, 1996
"... This paper deals with aspects of my doctoral dissertation 1 ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
This paper deals with aspects of my doctoral dissertation 1
A ContextualHierarchical Approach to Truth and the Liar Paradox
 Journal of Philosophical Logic
, 2004
"... This paper presents an approach to truth and the Liar paradox which combines elements of context dependence and hierarchy. This approach is developed formally, using the techniques of model theory in admissible sets. Special attention is paid to showing how starting with some ideas about context dra ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
This paper presents an approach to truth and the Liar paradox which combines elements of context dependence and hierarchy. This approach is developed formally, using the techniques of model theory in admissible sets. Special attention is paid to showing how starting with some ideas about context drawn from linguistics and philosophy of language, we can see the Liar sentence to be context dependent. Once this context dependence is properly understood, it is argued, a hierarchical structure emerges which is neither ad hoc nor unnatural. Your bait of falsehood takes this carp of truth: And thus do we of wisdom and of reach, With windlasses, and with assays of bias, By indirections find directions out: Hamlet II.i.68–71 It is a perennial idea in the study of the Liar paradox, from Tarski [56] onwards, that its solution requires some kind of hierarchy. More recent, but
HIGHER RANDOMNESS NOTIONS AND THEIR LOWNESS PROPERTIES
, 2007
"... Abstract. We study randomness notions given by higher recursion theory, establishing the relationships Π 1 1randomness ⊂ Π 1 1MartinLöf randomness ⊂ ∆ 1 1randomness = ∆ 1 1MartinLöf randomness. We characterize the set of reals that are low for ∆ 1 1 randomness as precisely those that are ∆ 1 ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Abstract. We study randomness notions given by higher recursion theory, establishing the relationships Π 1 1randomness ⊂ Π 1 1MartinLöf randomness ⊂ ∆ 1 1randomness = ∆ 1 1MartinLöf randomness. We characterize the set of reals that are low for ∆ 1 1 randomness as precisely those that are ∆ 1 1traceable. We prove that there is a perfect set of such reals. 1.
Centre for Discrete Mathematics and
, 2008
"... Abstract. A real x is ∆ 1 1Kurtz random (Π 1 1Kurtz random) if it in no closed null ∆ 1 1 set (Π 1 1 set). We show that there is a cone of Π 1 1Kurtz random hyperdegrees. We characterize lowness for ∆ 1 1Kurtz randomness by being ∆ 1 1dominated and ∆ 1 1semitraceable. 1. ..."
Abstract
 Add to MetaCart
Abstract. A real x is ∆ 1 1Kurtz random (Π 1 1Kurtz random) if it in no closed null ∆ 1 1 set (Π 1 1 set). We show that there is a cone of Π 1 1Kurtz random hyperdegrees. We characterize lowness for ∆ 1 1Kurtz randomness by being ∆ 1 1dominated and ∆ 1 1semitraceable. 1.
HIGHER KURTZ RANDOMNESS
"... Abstract. A real x is ∆ 1 1Kurtz random (Π 1 1Kurtz random) if it is in no closed null ∆ 1 1 set (Π 1 1 set). We show that there is a cone of Π 1 1Kurtz random hyperdegrees. We characterize lowness for ∆ 1 1Kurtz randomness as being ∆ 1 1dominated and ∆ 1 1semitraceable. 1. ..."
Abstract
 Add to MetaCart
Abstract. A real x is ∆ 1 1Kurtz random (Π 1 1Kurtz random) if it is in no closed null ∆ 1 1 set (Π 1 1 set). We show that there is a cone of Π 1 1Kurtz random hyperdegrees. We characterize lowness for ∆ 1 1Kurtz randomness as being ∆ 1 1dominated and ∆ 1 1semitraceable. 1.
ON SECONDORDER CHARACTERIZABILITY
"... Abstract. We investigate the extent of second order characterizable structures by extending Shelah’s Main Gap dichotomy to second order logic. For this end we consider a countable complete first order theory T. We show that all sufficiently large models of T have a characterization up to isomorphism ..."
Abstract
 Add to MetaCart
Abstract. We investigate the extent of second order characterizable structures by extending Shelah’s Main Gap dichotomy to second order logic. For this end we consider a countable complete first order theory T. We show that all sufficiently large models of T have a characterization up to isomorphism in the extension of second order logic obtained by adding a little bit of infinitary logic if and only if T is shallow superstable with NDOP and NOTOP. Our result relies on cardinal arithmetic assumptions. Under weaker assumptions we get consistency results or alternatively results about second order logic with Henkin semantics. Contents
COHEN AND SET THEORY
"... Abstract. We discuss the work of Paul Cohen in set theory and its influence, especially the background, discovery, development of forcing. Paul Joseph Cohen (1934–2007) in 1963 established the independence of the Axiom of Choice (AC) from ZF and the independence of the Continuum Hypothesis (CH) from ..."
Abstract
 Add to MetaCart
Abstract. We discuss the work of Paul Cohen in set theory and its influence, especially the background, discovery, development of forcing. Paul Joseph Cohen (1934–2007) in 1963 established the independence of the Axiom of Choice (AC) from ZF and the independence of the Continuum Hypothesis (CH) from ZFC. That is, he established that Con(ZF) implies Con(ZF+¬AC) and Con(ZFC) implies Con(ZFC+¬CH). Already prominent as an analyst, Cohen had ventured into set theory with fresh eyes and an openmindedness about possibilities. These results delimited ZF and ZFC in terms of the two fundamental issues at the beginnings of set theory. But beyond that, Cohen’s proofs were the inaugural examples of a new technique, forcing, which was to become a remarkably general and flexible method for extending models of set theory. Forcing has strong intuitive underpinnings and reinforces the notion of set as given by the firstorder ZF axioms with conspicuous uses of Replacement and Foundation. If Gödel’s construction of L had launched set theory as a distinctive field of mathematics, then Cohen’s forcing began its transformation into a modern, sophisticated one. The extent and breadth of the expansion of set theory henceforth dwarfed all that came before, both in terms of the numbers of people involved and the results established. With clear intimations of a new and concrete way of building models, set theorists rushed in and with forcing were soon establishing a cornucopia of relative consistency results, truths in a wider sense, with some illuminating classical problems of mathematics. Soon, ZFC became quite unlike Euclidean geometry and much like group theory, with a wide range of models of set theory being investigated for their own sake. Set theory had undergone a seachange, and with the subject so enriched, it is difficult to convey the strangeness of it. Received April 24, 2008. This is the full text of an invited address given at the annual meeting of the Association
Professor Kazuyuki Tanaka for organizing the school and inviting him to Sendai.
"... We consider implicit definability over the natural number system N,+,×,=. We present a new proofof two theorems ofLeo Harrington. The first theorem says that there exist implicitly definable subsets of N whicharenotexplicitly definablefromeachother. The secondtheorem says that there exists a subset ..."
Abstract
 Add to MetaCart
We consider implicit definability over the natural number system N,+,×,=. We present a new proofof two theorems ofLeo Harrington. The first theorem says that there exist implicitly definable subsets of N whicharenotexplicitly definablefromeachother. The secondtheorem says that there exists a subset of N which is not implicitly definable but belongs to a countable, explicitly definable set of subsets of N. Previous proofs of these theorems have used finite or infiniteinjury priority constructions. Our new proof is easier in that it uses only a nonpriority oracle construction, adapted from the standard proof of