The only reason I can see to prefer it is if you're not comfortable with transfinite induction.
12.
The well-ordering of the values of single parameters is provided by the inductive hypothesis of the transfinite induction.
13.
One common use of the Borel hierarchy is to prove facts about the Borel sets using transfinite induction on rank.
14.
We proceed by transfinite induction on ? to show that ? is ?-inaccessible for any ? d " ?.
15.
Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a limit process involving transfinite induction.
16.
Ordinal analysis was originated by Genzten, who proved the consistency of Peano Arithmetic using transfinite induction up to ordinal ? 0.
17.
It then follows by transfinite induction that there is one and only one function satisfying the recursion formula up to and including ?.
18.
Trying to create a general theory, Denjoy used transfinite induction over the possible types of singularities, which made the definition quite complicated.
19.
Such a definition is normally said to be by transfinite recursion the proof that the result is well-defined uses transfinite induction.
20.
In 1936, Gerhard Gentzen gave a proof of the consistency of Peano's axioms, using transfinite induction up to an ordinal called ? 0.
