Limit cardinal

This means that one cannot "reach" λ from another cardinal by repeated successor operations.

A cardinal λ is a strong limit cardinal if λ cannot be reached by repeated powerset operations.

(aleph-naught), is a strong limit cardinal, and hence also a weak limit cardinal.

One way to construct limit cardinals is via the union operation:

is a weak limit cardinal, defined as the union of all the alephs before it; and in general

The ב operation can be used to obtain strong limit cardinals.

Thus there are arbitrarily large strong limit cardinals.

If the axiom of choice holds, every cardinal number has an initial ordinal.

then the cardinal number is of the form

is a weak limit cardinal.

Although the ordinal subscript tells us whether a cardinal is a weak limit, it does not tell us whether a cardinal is a strong limit.

For example, ZFC proves that

is a weak limit cardinal, but neither proves nor disproves that

is a strong limit cardinal (Hrbacek and Jech 1999:168).

The generalized continuum hypothesis states that

for every infinite cardinal κ.

Under this hypothesis, the notions of weak and strong limit cardinals coincide.

The preceding defines a notion of "inaccessibility": we are dealing with cases where it is no longer enough to do finitely many iterations of the successor and powerset operations; hence the phrase "cannot be reached" in both of the intuitive definitions above.

But the "union operation" always provides another way of "accessing" these cardinals (and indeed, such is the case of limit ordinals as well).

Stronger notions of inaccessibility can be defined using cofinality.

The preceding examples both are singular cardinals of cofinality ω and hence they are not inaccessible.

Standard Zermelo–Fraenkel set theory with the axiom of choice (ZFC) cannot even prove the consistency of the existence of an inaccessible cardinal of either kind above

, due to Gödel's incompleteness theorem.

These form the first in a hierarchy of large cardinals.