Large cardinal

A large cardinal is an infinite cardinal whose existence is asserted by a strong set-theoretic axiom beyond what standard axioms can establish. Such cardinals are organized in a hierarchy by the strength of their defining properties.

Cardinal number: A cardinal number measures the size of a set, including infinite sets. Large-cardinal properties are imposed on cardinals to distinguish them from ordinary infinite sizes.
Elementary embedding: A map between structures that preserves the truth of every formula in their language. Many large cardinals are characterized by embeddings of the set-theoretic universe into a larger structure.
Inaccessible cardinal: An uncountable regular strong limit cardinal. Inaccessibility is a basic large-cardinal property and marks a natural early stage in the hierarchy.
Constructible universe: The inner model L, built from sets definable in stages from earlier levels. L satisfies choice and has no large cardinals, illustrating how restrictive a canonical inner model can be.
Determinacy: The property that one of two players has a winning strategy in a specified infinite game. Strong large-cardinal assumptions imply determinacy results for increasingly broad classes of games.
Axiom of infinity: An axiom of set theory asserting that an infinite set exists. It guarantees infinite sets, but does not establish the stronger cardinal properties at issue here.
Measurable cardinal: An uncountable cardinal carrying a nonprincipal, κ-complete ultrafilter. It is a canonical large cardinal whose ultrafilter yields an elementary embedding.
Weakly compact cardinal: An uncountable cardinal with strong partition and compactness properties. Weak compactness combines reflection and combinatorial compactness in a classic large-cardinal example.
Gödel's incompleteness theorems: Theorems showing that sufficiently expressive consistent formal systems cannot prove every truth or their own consistency. They help explain why existence claims stronger than a theory's axioms may remain undecidable within it.
Descriptive set theory: The study of definable sets of real numbers and related structures. Large cardinals establish regularity and determinacy properties for definable sets of reals.

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