Universal functions

• Autorzy: Paul B. Larson , Arnold W. Miller , Juris Steprāns , William A. R. Weiss
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 03E50: Continuum hypothesis and Martin's axiom
• 03E15: Descriptive set theory
• 03E35: Consistency and independence results

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2014
• Tom: 227
• Numer: 3
• Strony: 197-245

Daty

wydano

2014

Twórcy

autor

Paul B. Larson

• Department of Mathematics, Miami University, Oxford, OH 45056, U.S.A.

autor

Arnold W. Miller

• Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706-1388, U.S.A.

autor

Juris Steprāns

• Department of Mathematics, York University, 4700 Keele Street, Toronto, ON, Canada M3J 1P3

autor

William A. R. Weiss

• Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 3G3

Identyfikatory

DOI

10.4064/fm227-3-1