Journal of Symbolic Logic

On the Ranked Points of A $\Pi^0_1$ Set

Douglas Cenzer and Rick L. Smith

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This paper continues joint work of the authors with P. Clote, R. Soare and S. Wainer (Annals of Pure and Applied Logic, vol. 31 (1986), pp. 145--163). An element $x$ of the Cantor space $2^\omega$ is said have rank $\alpha$ in the closed set $P$ if $x$ is in $D^\alpha(P)\backslash D^{\alpha + 1}(P)$, where $D^\alpha$ is the iterated Cantor-Bendixson derivative. The rank of $x$ is defined to be the least $\alpha$ such that $x$ has rank $\alpha$ in some $\Pi^0_1$ set. The main result of the five-author paper is that for any recursive ordinal $\lambda + n$ (where $\lambda$ is a limit and $n$ is finite), there is a point with rank $\lambda + n$ which is Turing equivalent to $O^{(\lambda + 2n)}$. All ranked points constructed in that paper are $\Pi^0_2$ singletons. We now construct a ranked point which is not a $\Pi^0_2$ singleton. In the previous paper the points of high rank were also of high hyperarithmetic degree. We now construct $\triangle^0_2$ points with arbitrarily high rank. We also show that every nonrecursive RE point is Turing equivalent to an RE point of rank one and that every nonrecursive $\triangle^0_2$ point is Turing equivalent to a hyperimmune point of rank one. We relate Clote's notion of the height of a $\Pi^0_1$ singleton in the Baire space with the notion of rank. Finally, we show that every hyperimmune point $x$ is Turing equivalent to a point which is not ranked.

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J. Symbolic Logic, Volume 54, Issue 3 (1989), 975-991.

First available in Project Euclid: 6 July 2007

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Cenzer, Douglas; Smith, Rick L. On the Ranked Points of A $\Pi^0_1$ Set. J. Symbolic Logic 54 (1989), no. 3, 975--991.

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