In fact, Scott showed that (in contrast to the small large cardinals) measurable cardinals cannot exist in Gödel's constructible universe. Let us be precise about this. Let V=L be the statement that asserts that all sets are constructible. Then for each small large cardinal axiom φ (to be precise, those listed above) if the theory ZFC+φ is consistent then so is the theory ZFC+φ+V=L. In contrast, the theory ZFC + “There is a measurable cardinal” proves ¬V=L. This may seem somewhat counterintu