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We introduce the notion of {\cal C}-system of filters, generalizing the standard definitions of both extenders and towers of normal ideals. This provides a framework to develop the theory of extenders and towers in a more general and concise way. In this framework we investigate the topic of definability of generic large cardinals properties.
From large cardinals we show the consistency of normal, fine, κ-complete λ-dense ideals on {{\cal P}_\kappa }\left( \lambda \right) for successor κ. We explore the interplay between dense ideals, cardinal arithmetic, and squares, answering some open questions of Foreman.
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