When going from finite to infinite sets, properties arise that are impossible for finite sets. For example, an infinite set has bijections to proper subsets, and to the Cartesian product of the set with itself.
Now I wonder if for sets whose cardinality is $\ge$ an inaccessible cardinal, there are also new properties arising that are true for all such sets, but are not possible for smaller sets.
Now obviously there are new properties like “has a cardinality larger than an inaccessible cardinal” but I'm interested in things that can be formulated without reference to large cardinals (just as my two examples of infinite set properties nowhere reference infinity).
Clarification: The properties I'm after are set properties that under the assumption of ZFC + large cardinal, are equivalent to the property “the set's cardinality is greater or equal to an inaccessible cardinal”.