cardinal numbers, and is regular.

is the next cardinal number after the sequence , , , , and so on. Its initial ordinal is the limit of the sequence , , , , and so on, which has order type , so is singular, and so is . Assuming the axiom of choice, is the first infinite cardinal that is singular (the first infinite ordinal that is singular is , and the first infinite limit ordinal that is singular is ). Proving the existence of singular cardinals requires the axiom of replacement,