In mathematical analysis, a strong measure zero set[1] is a subset A of the real line with the following property: (Here |In| denotes the length of the interval In.)
[2] Borel's conjecture[1] states that every strong measure zero set is countable.
Sierpiński proved in 1928 that the continuum hypothesis (which is now also known to be independent of ZFC) implies the existence of uncountable strong measure zero sets.
[3] In 1976 Laver used a method of forcing to construct a model of ZFC in which Borel's conjecture holds.
The following characterization of strong measure zero sets was proved in 1973: This result establishes a connection to the notion of strongly meagre set, defined as follows: The dual Borel conjecture states that every strongly meagre set is countable.