Editing Transfinite induction (section)

==Induction by cases==
Let <math>P(\alpha)</math> be a [[Property (mathematics)|property]] defined for all ordinals <math>\alpha</math>. Suppose that whenever <math>P(\beta)</math> is true for all  <math>\beta < \alpha</math>, then <math>P(\alpha)</math> is also true.<ref>It is not necessary here to assume separately that <math>P(0)</math> is true.  As there is no <math>\beta</math> less than 0, it is [[vacuously true]] that for all <math>\beta<0</math>, <math>P(\beta)</math> is true.</ref>  Then transfinite induction tells us that <math>P</math> is true for all ordinals.

Usually the proof is broken down into three cases:

* '''Zero case:''' Prove that <math>P(0)</math> is true.
* '''Successor case:''' Prove that for any [[successor ordinal]] <math>\alpha+1</math>, <math>P(\alpha+1)</math> follows from <math>P(\alpha)</math> (and, if necessary, <math>P(\beta)</math> for all <math>\beta < \alpha</math>).
* '''Limit case:''' Prove that for any [[limit ordinal]] <math>\lambda</math>, if <math>P(\beta)</math> holds for all <math>\beta < \lambda</math>, then <math>P(\lambda)</math>.

All three cases are identical except for the type of ordinal considered.  They do not formally need to be considered separately, but in practice the proofs are typically so different as to require separate presentations. Zero is sometimes considered a [[limit ordinal]] and then may sometimes be treated in proofs in the same case as limit ordinals.