Universal functions

作者：Larson Paul B^*; Miller Arnold W; Steprans Juris; Weiss William A R

来源：Fundamenta Mathematicae, 2014, 227(3): 197-245.

DOI：10.4064/fm227-3-1

• Borel function
• universal
• Martin's Axiom
• Baire class
• cardinality of the continuum
• Cohen real model

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. Sierpinski 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 <= alpha < omega(1) there is a universal function of class a but none of class beta < alpha. We show that it is consistent with ZFC that there is no universal 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.

• 出版日期2014