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Vaught's conjecture for theories of one unary operation

Fundamenta Mathematicae

On some properties of Hurewicz, Menger, and Rothberger

Fundamenta Mathematicae

Vitali sets and Hamel bases that are Marczewski measurable

Fundamenta Mathematicae

We give examples of a Vitali set and a Hamel basis which are Marczewski measurable and perfectly dense. The Vitali set example answers a question posed by Jack Brown. We also show there is a Marczewski null Hamel basis for the reals, although a Vitali set cannot be Marczewski null. The proof of the existence of a Marczewski null Hamel basis for the plane is easier than for the reals and we give it first. We show that there is no easy way to get a Marczewski null Hamel basis for the reals from one...

Measurability of functions with approximately continuous vertical sections and measurable horizontal sections

Colloquium Mathematicae

Fundamenta Mathematicae

A MAD (maximal almost disjoint) family is an infinite subset of the infinite subsets of ω = 0,1,2,... such that any two elements of intersect in a finite set and every infinite subset of ω meets some element of in an infinite set. A Q-set is an uncountable set of reals such that every subset is a relative ${G}_{\delta }$-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 $P\left(\omega \right)={2}^{\omega }$.

Categoricity without equality

Fundamenta Mathematicae

We study categoricity in power for reduced models of first order logic without equality.

Universal functions

Fundamenta Mathematicae

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...

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