This is a talk at the CUNY Set Theory Seminar on 4 April 2014.
In 1919 Borel introduced the notion of a strong measure zero set and stated what has become known as Borel’s conjecture (BC): every strong measure zero set of reals is countable. Sierpiński soon proved (1928) that CH implies the failure of BC. However, a proof for the consistency of BC with ZFC would have to wait for the development of more powerful tools. In 1976, Laver used an iterated forcing argument to produce a model of ZFC + BC. I will present an exposition of these classical results. Time permitting, I will sketch some analogous results for the dual Borel Conjecture, the category analogue of BC.
(It turned out time was not permitting.)