Conventional large cardinals have been codified to have a certain form--postulating class sized objects. Though these are well-understood to have "equivalent" statements in ZFC, they don't actually "live in V". One can stipulate some very similar objects that can be thought of as "generic" large cardinals. The equivalent ZFC versions of these objects can have small cardinalities. As a result they are directly relevant to questions such as the Continuum Hypothesis. Moreover, generic elementary embeddings have become an essential technique for extracting consequences of large cardinals involving sets of small cardinality.

This lecture will show that a broad class of generic elementary embeddings is equiconsistent with their analogous large cardinals. The results include equiconsistency results between combinatorial properties of the first few uncountable cardinals and huge cardinals.