A large cardinal property strengthens uncountability by imposing combinatorial, logical, measure-like, or elementary-embedding structure whose existence is not provable in ZFC if ZFC is consistent.
A strongly inaccessible cardinal is an uncountable regular strong limit cardinal.
An uncountable cardinal is weakly compact when every -satisfiable theory in an language having at most nonlogical symbols is satisfiable.
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In set theory, a large cardinal is a type of cardinal number that possesses certain strong and often large-scale properties, which typically extend beyond the standard axioms of set theory (like Zermelo-Fraenkel set theory with the Axiom of Choice, ZFC). Large cardinals are significant in the study of the foundations of mathematics because they often have implications for the consistency and structure of set theory. There are various kinds of large cardinals, each with different defining properties.