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The first order theories of the Medvedev and the Muchnik lattice
"... We show that the first order theories of the Medevdev lattice and the Muchnik lattice are both computably isomorphic to the third order theory of true arithmetic. 1 ..."
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We show that the first order theories of the Medevdev lattice and the Muchnik lattice are both computably isomorphic to the third order theory of true arithmetic. 1
Direct and local definitions of the Turing jump
, 2008
"... We show that there are 5 formulas in the language of the Turing degrees, D, ..."
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We show that there are 5 formulas in the language of the Turing degrees, D,
The nr.e. degrees: undecidability and Σ1 substructures
, 2012
"... We study the global properties of Dn, the Turing degrees of the nr.e. sets. In Theorem 1.5, we show that the first order theory of Dn is not decidable. In Theorem 1.6, we show that for any two n and m with n < m, Dn is not a Σ1substructure of Dm. 1 ..."
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We study the global properties of Dn, the Turing degrees of the nr.e. sets. In Theorem 1.5, we show that the first order theory of Dn is not decidable. In Theorem 1.6, we show that for any two n and m with n < m, Dn is not a Σ1substructure of Dm. 1
DEGREES AND REVERSE MATHEMATICS
, 2011
"... We investigate the complexity of mathematical problems from two perspectives: Medvedev degrees and reverse mathematics. In the Medvedev degrees, we calculate the complexity of its firstorder theory, and we also calculate the complexities of the firstorder theories of several related structures. We ..."
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We investigate the complexity of mathematical problems from two perspectives: Medvedev degrees and reverse mathematics. In the Medvedev degrees, we calculate the complexity of its firstorder theory, and we also calculate the complexities of the firstorder theories of several related structures. We characterize the joinirreducible Medvedev degrees and deduce several consequences for the interpretation of propositional logic in the Medvedev degrees. We equate the size of chains of Medvedev degrees with the size of chains of sets of reals under ⊆. In reverse mathematics, we analyze the strength of classical theorems of finite graph theory generalized to the countable. In particular, we consider Menger’s theorem, Birkhoff’s theorem, and unfriendly partitions. BIOGRAPHICAL SKETCH Paul was born on February 28, 1983 in Richland, Washington during the final episode Goodbye, Farewell and Amen of the popular television series M*A*S*H.