The well-ordering of the values of single parameters is provided by the inductive hypothesis of the transfinite induction.
32.
One common use of the Borel hierarchy is to prove facts about the Borel sets using transfinite induction on rank.
33.
Although Schultz was unsuccessful in this entry, the essay preceded certain features of Georg Cantors Theory of Transfinite numbers.
34.
The concept of transfinite numbers, however, was first considered by Indian Jaina mathematicians as far back as 400 BC.
35.
We proceed by transfinite induction on ? to show that ? is ?-inaccessible for any ? d " ?.
36.
Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a limit process involving transfinite induction.
37.
Ordinal analysis was originated by Genzten, who proved the consistency of Peano Arithmetic using transfinite induction up to ordinal ? 0.
38.
It contained Cantor's reply to his critics and showed how the transfinite numbers were a systematic extension of the natural numbers.
39.
Hence how one chooses to resolve Russell's paradox has direct implications on the status accorded to Cantor's transfinite arithmetic.
40.
This relationship between lightface sets and their indices is used to extend the lightface Borel hierarchy into the transfinite, via recursive ordinals.
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