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Large cardinals

Optional advanced material: inaccessible, measurable and larger cardinals, elementary embeddings of the universe, and the hierarchy of consistency strength that organises much of modern set theory.

math.found.large-cardinals

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Forcing and independence

Required prerequisites, all the way down: 24

You will learn to

Define inaccessible and measurable cardinals and show that V_κ satisfies ZFC when κ is inaccessible.
Explain why ZFC cannot prove that an inaccessible cardinal exists (if ZFC is consistent), and what 'consistency strength' means.
Describe how measurable cardinals correspond to non-trivial elementary embeddings j : V → M.

Definition

κ is (strongly) inaccessible if it is uncountable, regular, and 2^λ < κ for every λ < κ.
κ is measurable if there is a κ-complete non-principal ultrafilter on κ.
A theory T has at least the consistency strength of S if the consistency of T implies the consistency of S, provably in a weak base theory.

Theorem

If κ is inaccessible then V_κ is a model of ZFC.
Scott (1961): if a measurable cardinal exists, then V ≠ L.
Every measurable cardinal is inaccessible and is a limit of inaccessible cardinals.

Common misconceptions

Thinking large cardinals are merely very big numbers; their significance lies in consistency strength and their consequences, for instance for sets of reals.
Assuming large-cardinal axioms are known to be consistent; their consistency cannot be proved in ZFC and rests on mathematical evidence and judgement.

Optional discoveries

Higher categories Grothendieck universes, which correspond to inaccessible cardinals, are a common way of handling size issues in category theory.
← ZFC: axiomatic set theory
← Ordinals and cardinals
← Gödel's incompleteness theorems
← Descriptive set theory