  In this paper we present a quantitative analysis of the first and second
Borel-Cantelli Lemmas and of two of their generalisations: the
Erd\H{o}s-R\'enyi Theorem, and the Kochen-Stone Theorem. We will see that the
first three results have direct quantitative formulations, giving an explicit
relationship between quantitative formulations of the assumptions and the
conclusion. For the Kochen-Stone theorem, however, we can show that the
numerical bounds of a direct quantitative formulation are not computable in
general. Nonetheless, we obtain a quantitative formulation of the Kochen-Stone
Theorem using Tao's notion of metastability.
