= Large cardinal
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A <large cardinal> is an infinite <cardinal number> satisfying a strong additional property, such as inaccessibility, weak compactness, or carrying a highly complete measure. The phrase names a family of axioms rather than one numerical size threshold.
For example, a <strongly inaccessible cardinal> is an uncountable regular strong limit cardinal, whereas a <measurable cardinal> carries a nonprincipal highly complete <ultrafilter>. Their significance is structural: they support strong reflection and compactness principles and organize consistency strengths in <set theory>. Merely taking a larger <successor cardinal> does not ensure any of these properties.
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