Sujet : Re: The set of necessary FISONs
De : FTR (at) *nospam* nomail.afraid.org (FromTheRafters)
Groupes : sci.mathDate : 27. Feb 2025, 19:43:14
Autres entêtes
Organisation : Peripheral Visions
Message-ID : <vpqbo7$37r8k$1@dont-email.me>
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on 2/26/2025, WM supposed :
On 26.02.2025 15:42, FromTheRafters wrote:
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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Wrong. Zermelo began to work on the problems of set theory under Hilbert's influence and in 1902 published his first work concerning the addition of transfinite cardinals. His ω is the first transfinite ordinal.
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Finite "Mathematical Induction" but not infinite "Transfinite Induction" so Cantor likely used von Neumann's construction.
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Nonsense.
AI Overview
Cantor did not use either Zermelo or von Neumann's construction of natural numbers for transfinite induction, as his work on transfinite numbers predated both of their formalizations; he developed the concept of transfinite ordinals without the modern set-theoretic framework, and his construction would be considered closer to the concept of von Neumann ordinals in terms of its underlying idea of well-ordered sets, but not the exact technical details of the von Neumann definition.
Key points:
Cantor's work came first:
Cantor developed the concept of transfinite numbers before Zermelo and von Neumann axiomatized set theory, so he did not use their specific constructions of natural numbers.
Conceptual similarities:
While the details differed, Cantor's idea of transfinite ordinals conceptually aligns more with the later von Neumann construction, where each ordinal is a set containing all smaller ordinals.
Modern formalization:
Von Neumann later formalized the concept of transfinite ordinals within set theory, providing a robust framework for using transfinite induction.
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