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Topic: § 518 What did Fraenkel wish to express with his st
ory of Tristram Shandy?

Replies: 3   Last Post: Jun 20, 2014 8:08 PM

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Posts: 18,076
Registered: 1/29/05
§ 518 What did Fraenkel wish to express with his st
ory of Tristram Shandy?

Posted: Jun 20, 2014 6:20 AM
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The appearance that a set, so to speak, can "contain equally many elements" as a proper subset is in a certain contrast {{that is, so to say, an understatement}} with the well-known theorem: The whole is always larger than a part of it. This apparent contrast, already clearly recognized by Galilei, has historically been an essential obstacle to the admission of the notation of actual infinity, because it seemed to discredit the infinite sets possessing such a paradoxical property. In reality, however, this theorem of the whole and its part had been proven only in the domain of the finite, and there was no reason to expect, that it would maintain its validity in the giant step that leads from the finite to the infinite {{let alone any reason to accept the contrary}}.
Footnote: Even more paradoxical appears the equivalence between two infinite sets of apparently very different perimeter, if it is seemingly transferred into the practical life. The uncomfortable feeling occuring in this case disappears if one realizes that this reality is only ostensible and that our perception is not adjusted to it. Well-known is so the story of Tristram Shandy ... {{cp. § 077 of
The uncomfortable feeling does not at all disappear when we realize that the natural numbers have the same well-order as the days or years of Tristram Shandy and that when enumerating the rational numbers always one settled task implies an infinity of further tasks. Always infiniteyl many natural and rational numbers remain unpaired and there is not the least proof of equinumerousity.}}

[Adolf A. Fraenkel: "Einleitung in die Mengenlehre" 3rd ed., Springer, Berlin (1928) p. 24]
Original German Text:,21,Folie 21

Regards, WM

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