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Embeddings into the Medvedev and Muchnik lattices of Π0 1 classes
 Archive for Mathematical Logic
, 2004
"... Archive for Mathematical Logic Let Pw and PM be the countable distributive lattices of Muchnik and Medvedev degrees of nonempty Π 0 1 subsets of 2 ω, under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are latticeembeddable below any nonzero ele ..."
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Cited by 22 (17 self)
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Archive for Mathematical Logic Let Pw and PM be the countable distributive lattices of Muchnik and Medvedev degrees of nonempty Π 0 1 subsets of 2 ω, under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are latticeembeddable below any nonzero element of Pw. We show that many countable distributive lattices are latticeembeddable below any nonzero element of PM. 1
An extension of the recursively enumerable Turing degrees
 Journal of the London Mathematical Society
, 2006
"... Consider the countable semilattice RT consisting of the recursively enumerable Turing degrees. Although RT is known to be structurally rich, a major source of frustration is that no specific, natural degrees in RT have been discovered, except the bottom and top degrees, 0 and 0 ′. In order to overco ..."
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Cited by 22 (16 self)
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Consider the countable semilattice RT consisting of the recursively enumerable Turing degrees. Although RT is known to be structurally rich, a major source of frustration is that no specific, natural degrees in RT have been discovered, except the bottom and top degrees, 0 and 0 ′. In order to overcome this difficulty, we embed RT into a larger degree structure which is better behaved. Namely, consider the countable distributive lattice Pw consisting of the weak degrees (also known as Muchnik degrees) of mass problems associated with nonempty Π 0 1 subsets of 2ω. It is known that Pw contains a bottom degree 0 and a top degree 1 and is structurally rich. Moreover, Pw contains many specific, natural degrees other than 0 and 1. In particular, we show that in Pw one has 0 < d < r1 < inf(r2, 1) < 1. Here, d is the weak degree of the diagonally nonrecursive functions, and rn is the weak degree of the nrandom reals. It is known that r1 can be characterized as the maximum weak degree ofaΠ 0 1 subset of 2ω of positive measure. We now show that inf(r2, 1) can be characterized as the maximum weak degree of a Π 0 1 subset of 2ω, the Turing upward closure of which is of positive measure. We exhibit a natural embedding of RT into Pw which is onetoone, preserves the semilattice structure of RT, carries 0 to 0, and carries 0 ′ to 1. Identifying RT with its image in Pw, we show that all of the degrees in RT except 0 and 1 are incomparable with the specific degrees d, r1, and inf(r2, 1) inPw. 1.
Π 0 1 sets and models of WKL0
"... We show that any two Medvedev complete Π 0 1 subsets of 2 ω are recursively homeomorphic. We obtain a Π 0 1 set Q ′ of countable coded ωmodels of WKL0 with a strong homogeneity property. We show that if G is a generic element of Q ′ , then the ωmodel of WKL0 coded by G satisfies ∀X∀Y (if X is de ..."
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Cited by 8 (5 self)
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We show that any two Medvedev complete Π 0 1 subsets of 2 ω are recursively homeomorphic. We obtain a Π 0 1 set Q ′ of countable coded ωmodels of WKL0 with a strong homogeneity property. We show that if G is a generic element of Q ′ , then the ωmodel of WKL0 coded by G satisfies ∀X∀Y (if X is definable from Y, then X is Turing reducible to Y). We use a result of Kučera to refute some plausible conjectures concerning ωmodels of WKL0. We generalize our results to nonωmodels of WKL0. We discuss the significance of our results for foundations of mathematics.
Mass problems and measuretheoretic regularity
, 2009
"... Research supported by NSF grants DMS0600823 and DMS0652637. ..."
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Cited by 4 (3 self)
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Research supported by NSF grants DMS0600823 and DMS0652637.