Ripheus23

In the standard model of set theory, an inaccessible cardinal is loosely defined as one that cannot be reached from smaller cardinals "by the usual operations of transfinite arithmetic." Aleph-1 can be reached from aleph-zero by the successor operation and is therefore not inaccessible. Aleph-omega can be expressed as the sum of all the cardinals smaller than itself, so it is not inaccessible. Depending on how one evaluates the powerset operation's outputs, many other small aleph-numbers are not inaccessible.

In the hyperoperational model of transfinite arithmetic, however, a much more general and perhaps exact definition of inaccessibility can be supplied. Here, we say that a cardinal K is inaccessible if it cannot be reached from smaller cardinals via hyperoperations indexed by ordinals smaller than the initial ordinal of K. For example, suppose k is < K. Have the initial ordinal of K be O. Then K is inaccessible if no operation k ↑a x goes to K for a < O. So cardinals are inaccessible not in an absolute sense as such, but only relative to different levels of the hyperoperator sequence in transfinite space.