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Foundations of mathematics is the study of the most basic concepts and logical structure of mathematics, with an eye to the unity of human knowledge. Almost all of the problems studied in this book are motivated by an overriding foundational question: What are the appropriate axioms for mathematics? Through a series of case studies, these axioms are(More)
to thank Stephen Binns for discussing these topics with me. Abstract A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if every member of Q Turing computes a member of P. We say that P is strongly reducible to Q if every member of Q Turing computes a member of P via a fixed Turing functional. The(More)
A Turing degree a is said to be almost everywhere dominating if, for almost all X ∈ 2 ω with respect to the " fair coin " probability measure on 2 ω , and for all g : ω → ω Turing reducible to X, there exists f : ω → ω of Turing degree a which dominates g. We study the problem of characterizing the almost everywhere dominating Turing degrees and other,(More)
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Let Pw and PM be the countable distributive lattices of Muchnik and Medvedev degrees of non-empty Π 0 1 subsets of 2 ω , under Muchnik and Medvedev reducibility, respectively. We show that all countable distribu-tive lattices are lattice-embeddable below any non-zero element of Pw. We show that many countable distributive lattices are lattice-embeddable(More)
A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let Pw be the lattice of weak degrees of mass problems associated with nonempty Π 0 1(More)
Continuing the investigations of X. Yu and others, we study the role of set existence axioms in classical Lebesgue measure theory. We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA 0. We show that several well-known measure-theoretic propositions including the Vitali Covering Theorem are equivalent to WWKL over(More)