The Higher Infinite is the realm of the Large Cardinals, namely very large infinite cardinal numbers with strong combinatorial properties and whose existence cannot be proved from the standard Zermelo-Fraenkel axioms of set theory with Choice (ZFC). Large cardinal axioms of set theory assert the existence of such cardinals, which form a hierarchy that measures the consistency strength of mathematical theories. In this talk we will present some results, some old as well as some recent ones, that point to a unified view of the large cardinal hierarchy in terms of symmetry and self-similarity of the set-theoretic universe.

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