## Journal of Symbolic Logic

### Nonuniformization Results for the Projective Hierarchy

#### Abstract

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

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

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183743672

Mathematical Reviews number (MathSciNet)
MR1133100

Zentralblatt MATH identifier
0736.03016

JSTOR