Journal of Symbolic Logic

Monotone Reducibility and the Family of Infinite Sets

Douglas Cenzer

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Abstract

Let $A$ and $B$ be subsets of the space $2^N$ of sets of natural numbers. $A$ is said to be Wadge reducible to $B$ if there is a continuous map $\Phi$ from $2^N$ into $2^N$ such that $A = \Phi^{-1} (B); A$ is said to be monotone reducible to $B$ if in addition the map $\Phi$ is monotone, that is, $a \subset b$ implies $\Phi (a) \subset \Phi(b)$. The set $A$ is said to be monotone if $a \in A$ and $a \subset b$ imply $b \in A$. For monotone sets, it is shown that, as for Wadge reducibility, sets low in the arithmetical hierarchy are nicely ordered. The $\triangle^0_1$ sets are all reducible to the ($\Sigma^0_1$ but not $\triangle^0_1$) sets, which are in turn all reducible to the strictly $\triangle^0_2$ sets, which are all in turn reducible to the strictly $\Sigma^0_2$ sets. In addition, the nontrivial $\Sigma^0_n$ sets all have the same degree for $n \leq 2$. For Wadge reducibility, these results extend throughout the Borel hierarchy. In contrast, we give two natural strictly $\Pi^0_2$ monotone sets which have different monotone degrees. We show that every $\Sigma^0_2$ monotone set is actually $\Sigma^0_2$ positive. We also consider reducibility for subsets of the space of compact subsets of $2^N$. This leads to the result that the finitely iterated Cantor-Bendixson derivative $D^n$ is a Borel map of class exactly $2n$, which answers a question of Kuratowski.

Article information

Source
J. Symbolic Logic, Volume 49, Issue 3 (1984), 774-782.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183741617

Mathematical Reviews number (MathSciNet)
MR758928

Zentralblatt MATH identifier
0573.54030

JSTOR
links.jstor.org

Citation

Cenzer, Douglas. Monotone Reducibility and the Family of Infinite Sets. J. Symbolic Logic 49 (1984), no. 3, 774--782. https://projecteuclid.org/euclid.jsl/1183741617


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