Universal functions

Paul B. Larson ¹   ; Arnold W. Miller ²   ; Juris Steprāns ³   ; William A. R. Weiss ⁴

¹ Department of Mathematics Miami University Oxford, OH 45056, U.S.A.
² Department of Mathematics University of Wisconsin-Madison Van Vleck Hall 480 Lincoln Drive Madison, WI 53706-1388, U.S.A.
³ Department of Mathematics York University 4700 Keele Street Toronto, ON, Canada M3J 1P3
⁴ Department of Mathematics University of Toronto Toronto, ON, Canada M5S 3G3

Fundamenta Mathematicae, Tome 227 (2014) no. 3, pp. 197-245

Résumé

A function of two variables $F(x,y)$ is universal if for every function $G(x,y)$ there exist functions $h(x)$ and $k(y)$ such that $$G(x,y)=F(h(x),k(y))$$ for all $x,y$. Sierpiński showed that assuming the Continuum Hypothesis there exists a Borel function $F(x,y)$ which is universal. Assuming Martin's Axiom there is a universal function of Baire class 2. A universal function cannot be of Baire class 1. Here we show that it is consistent that for each $\alpha $ with $2\leq \alpha \omega _1$ there is a universal function of class $\alpha $ but none of class $\beta \alpha $. We show that it is consistent with ZFC that there is no universal function (Borel or not) on the reals, and we show that it is consistent that there is a universal function but no Borel universal function. We also prove some results concerning higher-arity universal functions. For example, the existence of an $F$ such that for every $G$ there are $h_1,h_2,h_3$ such that for all $x,y,z$, $$G(x,y,z)=F(h_1(x),h_2(y),h_3(z))$$ is equivalent to the existence of a binary universal $F$, however the existence of an $F$ such that for every $G$ there are $h_1,h_2,h_3$ such that for all $x,y,z$, $$G(x,y,z)=F(h_1(x,y),h_2(x,z),h_3(y,z))$$ follows from a binary universal $F$ but is strictly weaker.

Détail

DOI : 10.4064/fm227-3-1

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