Successor cardinal
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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 Infinity-style). 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
- <math>\kappa^+ = \inf \{ \lambda \in ON \ |\ \kappa < \lambda \}<math>
That the set above is nonempty follows from Hartogs' theorem, which says for a well-orderable cardinal, we can construct a larger one. The minimum actually exists because the ordinals are well-ordered. 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:
- <math>\aleph_0 = \omega<math>
- <math>\aleph_{\alpha+1} = \aleph_{\alpha}^+<math>
and for λ an infinite limit ordinal,
- <math>\aleph_{\lambda} = \bigcup_{\beta < \lambda} \aleph_\beta<math>
It is clear that <math>\aleph_{\beta}<math> 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 <math>\aleph_{\lambda}<math> is a limit cardinal.