Successor cardinal
In the theory of
cardinal numbers, we can define a successor operation similar to that in the
ordinal numbers. This coincides with the ordinal successor operation for finite cardinals, but in the infinite case they diverge because every infinite ordinal and its successor have the same
cardinality (an easy
bijection can be set up between the two by simply sending the last element of the successor to 0, 0 to 1, etc., and fixing ω and all the elements above; Hotel Infinitystyle). Using the
von Neumann cardinal assignment and the
axiom of choice (AC), this successor operation is easy to define: for a cardinal number κ we have
That the set above is nonempty follows from Hartogs' theorem, which says for a wellorderable cardinal, we can construct a larger one. The minimum actually exists because the ordinals are wellordered. It is therefore immediate that there is no cardinal number in between κ and κ
^{+}. A
successor cardinal is a cardinal which is κ
^{+} for some cardinal κ. In the infinite case, the successor operation skips over many ordinal numbers; in fact, every infinite cardinal is a
limit ordinal. Therefore, the successor operation on cardinals gains a lot of power in the infinite case (relative the ordinal successorship operation), and consequently the cardinal numbers are a very "sparse" subclass of the ordinals. We define the sequence of
alephs (via the
axiom of replacement) via this operation, through all the ordinal numbers as follows:

and for λ a limit ordinal,

It is clear that ℵ
_{β} for some
successor ordinal β is in fact a successor cardinal. Cardinals which are not successor cardinals are called
limit cardinals; and by the above definition, if λ is a limit ordinal, then ℵ
_{λ} is a limit cardinal.