Whether this is simply a linguistic convention, or something more, is a controversial point among distinct philosophical schools (see Motivations and epistemic status below).
Most working set theorists believe that the large cardinal axioms that are currently being considered are consistent with ZFC.
A remarkable observation about large cardinal axioms is that they appear to occur in strict linear order by consistency strength.
That is, no exception is known to the following: Given two large cardinal axioms A1 and A2, exactly one of three things happens: These are mutually exclusive, unless one of the theories in question is actually inconsistent.
(Without an accepted definition of large cardinal property, it is not subject to proof in the ordinary sense.)
Large cardinals are understood in the context of the von Neumann universe V, which is built up by transfinitely iterating the powerset operation, which collects together all subsets of a given set.
There are also realists who deny that ontological maximalism is a proper motivation, and even believe that large cardinal axioms are false.