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Exact pairs for the ideal of the $K$-trivial sequences in the Turing degrees

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Semantics and Syntax: A Legacy of Alan Turing

The $K$-trivial sets form an ideal in the Turing degrees, which is generated by its computably enumerable (c.e.) members and has an exact pair below the degree of the halting problem. The question of whether it has an exact pair in the c.e. degrees was first raised in a published list of questions in the Bulletin of Symbolic Logic in 2006 by Miller and Nies and later in Nies’ book on computability and randomness. Moreover it was featured in several conferences in algorithmic randomness, since 2005.

We give a negative answer to this question. In fact, we show the following stronger statement in the c.e. degrees. There exists a $K$-trivial degree $mathbf{d}$ such that for all degrees $mathbf{a}, mathbf{b}$ which are not $K$-trivial and $mathbf{a}>mathbf{d}, mathbf{b}>mathbf{d}$ there exists a degree $mathbf{v}$ which is not $K$-trivial and $mathbf{a}>mathbf{v}, mathbf{b}>mathbf{v}$. This work sheds light to the question of the definability of the $K$-trivial degrees in the c.e. degrees.

This talk is part of the Isaac Newton Institute Seminar Series series.

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