Title: Generic Large Cardinals: New Axioms for Mathematics?

This article discusses various attempts at strengthening the axioms for mathematics, \emph{Zermelo-Fraenkel Set Theory with the Axiom of Choice} It focuses on a relatively recent collection of axioms, \emph{generic large cardinals}, their success at settling well known independent problems and their relations to other strengthenings of ZFC, such as large cardinals.

1991 Mathematics Subject Classification: 3,4,5,28

Keywords and Phrases: axioms, large cardinals, ideals, generic large cardinals

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