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A MAD Q-set

100%
Miller A.
Fundamenta Mathematicae
|
2003
|
tom 178
|
nr 3
271-281
EN
A MAD (maximal almost disjoint) family is an infinite subset 𝒜 of the infinite subsets of ω = {0,1,2,...} such that any two elements of 𝒜 intersect in a finite set and every infinite subset of ω meets some element of 𝒜 in an infinite set. A Q-set is an uncountable set of reals such that every subset is a relative $G_δ$-set. It is shown that it is relatively consistent with ZFC that there exists a MAD family which is also a Q-set in the topology it inherits as a subset of $P(ω) = 2^{ω}$.
2
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Categoricity without equality

63%
Keisler H., Miller A.
Fundamenta Mathematicae
|
2001
|
tom 170
|
nr 1-2
87-106
EN
We study categoricity in power for reduced models of first order logic without equality.
3
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Universal functions

51%
Larson P., Miller A., Steprāns J., Weiss W.
Fundamenta Mathematicae
|
2014
|
tom 227
|
nr 3
197-245
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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