Re: The set of necessary FISONs

WM pretended :

 Consider Cantor's first application of transfinite induction in order to determine the power function in the second number class [G. Cantor: "Beiträge zur Begründung der transfiniten Mengenlehre 2", Math. Annalen 49 (1897) § 18. Cantor, p. 336ff]. In English:
https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 52.
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he merely showed that a countable list could not contain every real number.

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He showed a lot more.

https://mathresearch.utsa.edu/wiki/index.php?title=Natural_Numbers:Postulates#Zermelo_ordinals
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Each natural number is then equal to the set containing just the natural number preceding it. This is the definition of Zermelo ordinals. Unlike von Neumann's construction, the Zermelo ordinals do not account for infinite ordinals.
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Finite "Mathematical Induction" but not infinite "Transfinite Induction" so Cantor likely used von Neumann's construction.