Journal of Symbolic Logic

Monotone Reducibility and the Family of Infinite Sets

Douglas Cenzer

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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.

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

First available in Project Euclid: 6 July 2007

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Cenzer, Douglas. Monotone Reducibility and the Family of Infinite Sets. J. Symbolic Logic 49 (1984), no. 3, 774--782.

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