Journal of Symbolic Logic

Nonuniformization Results for the Projective Hierarchy

Steve Jackson and R. Daniel Mauldin

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Let $X$ and $Y$ be uncountable Polish spaces. We show in ZF that there is a coanalytic subset $P$ of $X \times Y$ with countable sections which cannot be expressed as the union of countably many partial coanalytic, or even $\mathrm{PCA} = \Sigma^1_2$, graphs. If $X = Y = \omega^\omega, P$ may be taken to be $\Pi^1_1$. Assuming stronger set theoretic axioms, we identify the least pointclass such that any such coanalytic $P$ can be expressed as the union of countably many graphs in this pointclass. This last result is extended (under suitable hypotheses) to all levels of the projective hierarchy.

Article information

J. Symbolic Logic, Volume 56, Issue 2 (1991), 742-748.

First available in Project Euclid: 6 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 04A15
Secondary: 03E60: Determinacy principles 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05]

Coanalytic countable uniformiza game quantifier


Jackson, Steve; Mauldin, R. Daniel. Nonuniformization Results for the Projective Hierarchy. J. Symbolic Logic 56 (1991), no. 2, 742--748.

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