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Tytuł artykułu

Universal functions

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EN
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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.
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Twórcy
  • Department of Mathematics, Miami University, Oxford, OH 45056, U.S.A.
  • Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706-1388, U.S.A.
  • Department of Mathematics, York University, 4700 Keele Street, Toronto, ON, Canada M3J 1P3
  • Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 3G3
Bibliografia
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Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm227-3-1
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