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Degree structures: Local and global investigations
 Bulletin of Symbolic Logic
"... $1. Introduction. The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead. ..."
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$1. Introduction. The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead.
EMBEDDING JUMP UPPER SEMILATTICES INTO THE TURING DEGREES
"... We prove that every countable jump upper semilattice can be embedded in D, where a jump upper semilattice (jusl) is an upper semilattice endowed with a strictly increasing and monotone unary operator that we call jump, and D is the jusl of Turing degrees. As a corollary we get that the existential ..."
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We prove that every countable jump upper semilattice can be embedded in D, where a jump upper semilattice (jusl) is an upper semilattice endowed with a strictly increasing and monotone unary operator that we call jump, and D is the jusl of Turing degrees. As a corollary we get that the existential theory of 〈D, ≤T, ∨, ′ 〉 is decidable. We also prove that this result is not true about jusls with 0, by proving that not every quantifier free 1type of jusl with 0 is realized in D. On the other hand, we show that every quantifier free 1type of jump partial ordering (jpo) with 0 is realized in D. Moreover, we show that if every quantifier free type, p(x1,..., xn), of jpo with 0, which contains the formula x1 ≤ 0 (m) &... & xn ≤ 0 (m) for some m, is realized in D, then every every quantifier free type of jpo with 0 is realized in D. We also study the question of whether every jusl with the c.p.p. and size κ ≤ 2 ℵ0 is embeddable in D. We show that for κ = 2 ℵ0 the answer is no, and that for κ = ℵ1 it is independent of ZFC. (It is true if MA(κ) holds.)
A General Framework for Priority Arguments
 The Bulletin of Symbolic Logic
, 1995
"... this paper. ..."
The ∀∃ theory of D(≤, ∨, ′ ) is undecidable
 In Proceedings of Logic Colloquium
, 2003
"... We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable. ..."
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We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable.
An Almost Deep Degree
 J. Symbolic Logic
"... We show there is a nonrecursive r.e. set A such that if W is any low r.e. set, then the join W # A is also low. That is, A is "almost deep". This answers a question of Jockusch. The almost deep degrees form an definable ideal in the r.e. degrees (with jump.) 1 Introduction Bickford and Mills ..."
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We show there is a nonrecursive r.e. set A such that if W is any low r.e. set, then the join W # A is also low. That is, A is "almost deep". This answers a question of Jockusch. The almost deep degrees form an definable ideal in the r.e. degrees (with jump.) 1 Introduction Bickford and Mills in [1] defined an r.e. degree a to be deep in case, for all other r.e. degrees b, (b # a) # = b # . In other words, joining with a preserves the jump of every r.e. degree. They asked whether there are nonrecursive deep degrees. Part of the motivation for asking this question is an interest in finding definable ideals in the r.e. degrees. Such ideals can help in understanding the global structure of the r.e. degrees; in particular they can provide ways of understanding definability properties and automorphisms. We know of few # Partially supported by NSF Grant DMS963465. + Partially supported by NSF Grant DMS9796121. 1 nontrivial examples of such ideals. One example is the idea...
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