Is there any number higher than an inaccessible cardinal?

If [math]\kappa[/math] is an inaccessible cardinal, then [math]\kappa + 1[/math] is a larger ordinal number, and [math]\aleph_{\kappa + 1}[/math] is a larger cardinal number. Now, we should note that there may not be another inaccessible cardinal greater than a particular inaccessible. read more

Other authors use the definition that for any ordinal α, a cardinal κ is α-hyper-inaccessible if and only if κ is hyper-inaccessible and for every ordinal β < α, the set of β-hyper-inaccessibles less than κ is unbounded in κ. read more

If they do exist it is certainly the case that any inaccessible cardinal is larger than the smallest transfinite cardinal, [math]\aleph_0[/math], Aleph Null, the cardinality of the set of Natural numbers, [math]\{0,1,2,3,\dotsc\}[/math], and a plausible candidate for your informal term “infinite”. read more