When you're doing a careful proof by transfinite induction, or a construction by transfinite recursion, ordinarily you should probably have three parts to the proof : stage 0, successor stages, and limit stages.
12.
Continuing in this vein, one can define maps ? ? for progressively larger ordinals ? ( including, by this rarefied form of transfinite recursion, limit ordinals ), with progressively larger least fixed points ? ? + 1 ( 0 ).
13.
Because unless there's something I'm missing with this transfinite recursion, I'm still not sure I accept the existence of sufficiently large ordinal numbers ( though I promise I will as soon as I accept the well-ordering principle ).
14.
For infinite groups, one can continue the lower central series to infinite ordinal numbers via transfinite recursion : for a limit ordinal ?, define " G " ? = ) " { " G " ? : ? < ? }.
15.
It might appear that this is the non-constructive equivalent of the constructive definition using iterated exponentiation; but the two definitions are equally non-constructive at steps indexed by limit ordinals, which represent transfinite recursion of a higher order than taking the supremum of an exponential series.
16.
For an infinite group, the derived series need not terminate at a finite stage, and one can continue it to infinite ordinal numbers via transfinite recursion, thereby obtaining the "'transfinite derived series "', which eventually terminates at the perfect core of the group.
17.
By transfinite recursion on ?, we can use transfinite recursion on ? to define ? ( ?, ? ) = the smallest ordinal ? such that ? < ? and ? < ? and ? is not the value of ? for any smaller ? or for the same ? with a smaller ?.
18.
By transfinite recursion on ?, we can use transfinite recursion on ? to define ? ( ?, ? ) = the smallest ordinal ? such that ? < ? and ? < ? and ? is not the value of ? for any smaller ? or for the same ? with a smaller ?.
19.
Here is an example of definition by transfinite recursion on the ordinals ( more will be given later ) : define function " F " by letting " F " ( ? ) be the smallest ordinal not in the set } }, that is, the set consisting of all " F " ( ? ) for.
20.
For example, a construction by transfinite recursion frequently will not specify a " unique " value for " A " ? + 1, given the sequence up to ?, but will specify only a " condition " that " A " ? + 1 must satisfy, and argue that there is at least one set satisfying this condition.
