Universal functions

Paul B. Larson¹ ; Arnold W. Miller² ; Juris Steprāns³ ; William A. R. Weiss⁴

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

Keywords: function variables universal every function there exist functions sierpi ski showed assuming continuum hypothesis there exists borel function which universal assuming martins axiom there universal function baire class universal function cannot baire class here consistent each alpha leq alpha omega there universal function class alpha none class beta alpha consistent zfc there universal function borel reals consistent there universal function borel universal function prove results concerning higher arity universal functions example existence every there z equivalent existence binary universal however existence every there z follows binary universal strictly weaker