## Notre Dame Journal of Formal Logic

### Classifying the Branching Degrees in the Medvedev Lattice of $\Pi^0_1$ Classes

Christopher P. Alfeld

#### Abstract

A $\Pi^0_1$ class can be defined as the set of infinite paths through a computable tree. For classes $P$ and $Q$, say that $P$ is Medvedev reducible to $Q$, $P \leq_M Q$, if there is a computably continuous functional mapping $Q$ into $P$. Let $\mathcal{L}_M$ be the lattice of degrees formed by $\Pi^0_1$ subclasses of $2^\omega$ under the Medvedev reducibility. In "Non-branching degrees in the Medvedev lattice of $\Pi \sp{0}\sb{1}$ classes," I provided a characterization of nonbranching/branching and a classification of the nonbranching degrees. In this paper, I present a similar classification of the branching degrees. In particular, $P$ is separable if there is a clopen set $C$ such that $P \cap C \neq \emptyset \neq P \cap C^c$ and $P \cap C \perp_M P \cap C^c$. By the results in the first paper, separability is an invariant of a Medvedev degree and a degree is branching if and only if it contains a separable member. Further define $P$ to be hyperseparable if, for all such $C$, $P \cap C \perp_M P \cap C^c$ and totally separable if, for all $X,Y \in P$, $X \perp_T Y$. I will show that totally separable implies hyperseparable implies separable and that the reverse implications do not hold, that is, that these are three distinct types of branching degrees. Along the way I will show some related results and present a combinatorial framework for constructing $\Pi^0_1$ classes with priority arguments.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 49, Number 3 (2008), 227-243.

Dates
First available in Project Euclid: 15 July 2008

https://projecteuclid.org/euclid.ndjfl/1216152548

Digital Object Identifier
doi:10.1215/00294527-2008-009

Mathematical Reviews number (MathSciNet)
MR2428552

Zentralblatt MATH identifier
1157.03021

Subjects
Primary: 03D30: Other degrees and reducibilities

#### Citation

Alfeld, Christopher P. Classifying the Branching Degrees in the Medvedev Lattice of $\Pi^0_1$ Classes. Notre Dame J. Formal Logic 49 (2008), no. 3, 227--243. doi:10.1215/00294527-2008-009. https://projecteuclid.org/euclid.ndjfl/1216152548

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