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Universal functions

Volume 227 / 2014

Paul B. Larson, Arnold W. Miller, Juris Steprāns, William A. R. Weiss Fundamenta Mathematicae 227 (2014), 197-245 MSC: Primary 03E15; Secondary 03E35, 03E50. DOI: 10.4064/fm227-3-1

Abstract

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.

Authors

  • Paul B. LarsonDepartment of Mathematics
    Miami University
    Oxford, OH 45056, U.S.A.
    e-mail
  • Arnold W. MillerDepartment of Mathematics
    University of Wisconsin-Madison
    Van Vleck Hall
    480 Lincoln Drive
    Madison, WI 53706-1388, U.S.A.
    e-mail
  • Juris SteprānsDepartment of Mathematics
    York University
    4700 Keele Street
    Toronto, ON, Canada M3J 1P3
    e-mail
  • William A. R. WeissDepartment of Mathematics
    University of Toronto
    Toronto, ON, Canada M5S 3G3
    e-mail

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