Borel conjecture
The Borel conjecture states that every strong measure zero subset of the real line is countable. It is independent of the usual axioms of set theory.
Lebesgue measure: A measure assigning lengths, areas, and volumes to suitable sets, extending ordinary geometric measurement. Strong measure zero is a covering property distinct from having Lebesgue measure zero.
Null set: A measurable set has measure zero when it can be covered by intervals of arbitrarily small total length. Strong measure zero implies measure zero, but the converse fails.
Georg Cantor: Georg Cantor was a German mathematician who founded set theory and studied the sizes of infinite sets. Cantor posed the problem of whether strong measure zero sets must be countable.
Continuum hypothesis: The continuum hypothesis asserts that no cardinality lies strictly between the natural numbers and the real numbers. Unlike this cardinality assertion, the Borel conjecture restricts sets by a covering property.
Independence of the continuum hypothesis: The independence result shows that the continuum hypothesis can neither be proved nor refuted from Zermelo–Fraenkel set theory with choice, assuming consistency. It exemplifies the independence methods later used to analyze the Borel conjecture.
Baire category: A framework that classifies topological spaces and sets through notions of meagerness and genericity. It provides a topological counterpart to measure, clarifying what strong measure zero does not mean.
Rothberger property: A topological covering property requiring one set from each sequence of open covers to cover the space. For subsets of the real line, this property characterizes strong measure zero.
Émile Borel: Émile Borel was a French mathematician whose work helped establish modern measure theory. Borel formulated the conjecture now named for him.
Martin's axiom: Martin's axiom is a statement about partially ordered sets that extends consequences of the countable chain condition. Some versions of this axiom imply every strong measure zero set has size below the continuum, weaker than countability.
Zermelo–Fraenkel set theory: Zermelo–Fraenkel set theory is an axiomatic foundation for mathematics, commonly used with the axiom of choice. The conjecture’s independence is assessed relative to this standard foundation.