>>14142469They're sets that can't be reached through cardinal arithmetic.
The natural numbers are weakly inaccessible. The existence of strongly inaccessible cardinals (uncountable cardinals k such that for any cardinal a < k, 2^a < k) implies the consistency of ZFC and is therefore either inconsistent with or independent from ZFC. It is provable from ZFC however that if strongly inaccessibles do exist, they are either weak limits or regular.
n-inaccessible cardinals (where n is an ordinal) are inaccessible cardinals k such that for any m<n the set of m-inaccessible cardinals is unbounded in k. If the cardinality of n is k, then it's a hyperinaccessible (and from there we can define n-hyperinaccessibles and so on). Mahlo cardinals take it a step further such that they're always inaccessible, hyperinaccessible, hyperhyperinaccessible and so on.
ZFD (ZF + Determinacy) implies the existence of infinitely many Woodin cardinals and that all sets of reals are Measurable and *probably* Ramsey (technically an open question).
