Universal functions

Paul B. Larson, et al. "Universal functions." Fundamenta Mathematicae 227.3 (2014): 197-245. <http://eudml.org/doc/283259>.

@article{PaulB2014,
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 α with 2 ≤ α < ω₁ there is a universal function of class α but none of class β <α. 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 h₁,h₂,h₃ such that for all x,y,z, G(x,y,z) = F(h₁(x),h₂(y),h₃(z)) is equivalent to the existence of a binary universal F, however the existence of an F such that for every G there are h₁,h₂,h₃ such that for all x,y,z, G(x,y,z) = F(h₁(x,y),h₂(x,z),h₃(y,z)) follows from a binary universal F but is strictly weaker. },
author = {Paul B. Larson, Arnold W. Miller, Juris Steprāns, William A. R. Weiss},
journal = {Fundamenta Mathematicae},
keywords = {Borel function; universal; martin's axiom; Baire class; cardinality of the continuum; Cohen real model},
language = {eng},
number = {3},
pages = {197-245},
title = {Universal functions},
url = {http://eudml.org/doc/283259},
volume = {227},
year = {2014},
}

TY - JOUR
AU - Paul B. Larson
AU - Arnold W. Miller
AU - Juris Steprāns
AU - William A. R. Weiss
TI - Universal functions
JO - Fundamenta Mathematicae
PY - 2014
VL - 227
IS - 3
SP - 197
EP - 245
AB - 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 α with 2 ≤ α < ω₁ there is a universal function of class α but none of class β <α. 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 h₁,h₂,h₃ such that for all x,y,z, G(x,y,z) = F(h₁(x),h₂(y),h₃(z)) is equivalent to the existence of a binary universal F, however the existence of an F such that for every G there are h₁,h₂,h₃ such that for all x,y,z, G(x,y,z) = F(h₁(x,y),h₂(x,z),h₃(y,z)) follows from a binary universal F but is strictly weaker.
LA - eng
KW - Borel function; universal; martin's axiom; Baire class; cardinality of the continuum; Cohen real model
UR - http://eudml.org/doc/283259
ER -