Large Cardinals

Set Theory · Axiom Academy

Cardinals so large they transcend the reach of ZFC's constructive power The first level of large cardinals: a cardinal κ is inaccessible if: κ is a regular cardinal (not the union of fewer than κ sets each of size less than κ) κ is a strong limit cardinal (if λ < κ, then 2^λ < κ) Inaccessible cardinals "can't be reached from below"—you cannot construct κ by taking power sets and unions of smaller cardinals. A cardinal κ is measurable if there exists a κ-complete non-principal ultrafilter on κ. Equivalently, there is a "measure" on subsets of κ satisfying nice properties. Measurable cardinals are much larger than inaccessible cardinals. In fact, if κ is measurable, there are κ-many inaccessible cardinals below it! U is κ-complete: closed under intersections of fewer than κ sets U is non-principal: contains no singletons For every X ⊆ κ, either X ∈ U or κ \ X ∈ U Measurable cardinals have important connections to model theory through ultraproducts and elementary embeddings. 3. The Large Cardinal Hierarchy Large cardinals form a vast hierarchy of increasing strength. Moving up the hierarchy, we encounter: Weakly compact: A mild strengthening of inaccessibility Measurable: As described above Strong: Related to elementary embeddings that preserve more structure Woodin: Important for descriptive set theory Supercompact: Very powerful; imply strong reflection principles Huge, superhuge, and beyond: The hierarchy continues indefinitely!

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