Generic large cardinals: New axioms for mathematics?
This expository article surveys some prominent independence problems and discusses the various axioms (or axiom systems) which have been introduced to settle them: the axiom of constructibility, large cardinal hypotheses, AD, Martin's maximum, the generic large cardinal axioms. It focuses on the latter which assert the existence of large cardinal type elementary embeddings definable in forcing extensions of the universe. The advantage of generic elementary embeddings is that their critical point can be a small cardinal such as \(\omega_1\). Generic large cardinal axioms can often be reformulated as asserting the existence of ideals with strong properties, e.g., \(\kappa^+\)-saturated ideals on a regular cardinal \(\kappa\). The paper is an excellent introduction to the subject.
scientific article; zbMATH DE number 1184259
scientific article