Editing Successor cardinal

In [[set theory]], one can define a '''successor''' operation on [[cardinal number]]s in a similar way to the successor operation on the [[ordinal number]]s. The cardinal successor coincides with the ordinal successor for finite cardinals, but in the infinite case they diverge because every infinite ordinal and its successor have the same [[cardinality]] (a [[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; in the style of Hilbert's [[Hilbert's paradox of the Grand Hotel|Hotel Infinity]]). 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^+ = \left|\inf \{ \lambda \in \mathrm{ON} \ :\ \kappa < \left|\lambda\right| \}\right|</math> ,

where ON is the [[Class (set theory)|class]] of ordinals.  That is, the successor cardinal is the cardinality of the least ordinal into which a set of the given cardinality can be mapped one-to-one, but which cannot be mapped one-to-one back into that set.

That the set above is nonempty follows from  [[Hartogs number|Hartogs' theorem]], which says that for any [[well-order]]able cardinal, a larger such cardinal is constructible. 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''<sup>+</sup>. A '''successor cardinal''' is a cardinal that is ''κ''<sup>+</sup> 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 [[aleph number|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>

If ''β'' is a [[successor ordinal]], then <math>\aleph_{\beta}</math> is  a successor cardinal. Cardinals that are not successor cardinals are called [[limit cardinal]]s; and by the above definition, if ''λ'' is a limit ordinal, then <math>\aleph_{\lambda}</math> is a limit cardinal.

The standard definition above is restricted to the case when the cardinal can be well-ordered, i.e. is finite or an aleph.  Without the axiom of choice, there are cardinals that cannot be well-ordered.  Some mathematicians have defined the successor of such a cardinal as the cardinality of the least ordinal that cannot be mapped one-to-one into a set of the given cardinality.  That is:

:<math>\kappa^+ = \left|\inf \{ \lambda \in \mathrm{ON} \ :\ |\lambda| \nleq \kappa \}\right|</math>

which is the [[Hartogs number]] of ''κ''.

==See also==

*[[Cardinal assignment]]

==References==
{{refbegin}}
*[[Paul Halmos]], ''Naive set theory''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. {{isbn|0-387-90092-6}} (Springer-Verlag edition).
*[[Thomas Jech|Jech, Thomas]], 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''.  Springer.  {{isbn|3-540-44085-2}}.
*[[Kenneth Kunen|Kunen, Kenneth]], 1980. ''Set Theory: An Introduction to Independence Proofs''. Elsevier.  {{isbn|0-444-86839-9}}.
{{refend}}

{{Mathematical logic}}

[[Category:Cardinal numbers]]
[[Category:Set theory]]