Generic Large Cardinals and Systems of Filters

TL;DR

This paper introduces a generalized framework for large cardinal systems using $\

Contribution

It develops a unifying theory of extenders and towers of normal ideals through $\\mathcal{C}$-systems of filters, enabling broader analysis of generic large cardinal properties.

Findings

Unified framework for extenders and towers

Enhanced understanding of definability of large cardinals

Generalization of existing large cardinal concepts

Abstract

We introduce the notion of $\mathcal{C}$-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.