Sujet : Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
De : FTR (at) *nospam* nomail.afraid.org (FromTheRafters)
Groupes : sci.mathDate : 29. Nov 2024, 22:50:20
Autres entêtes
Organisation : Peripheral Visions
Message-ID : <vidcv3$18pdu$1@dont-email.me>
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WM wrote on 11/29/2024 :
On 29.11.2024 13:23, FromTheRafters wrote:
WM was thinking very hard :
On 29.11.2024 09:54, FromTheRafters wrote:
WM expressed precisely :
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But as long as infinitely many natnumbers have not left the endsegments, they stay inside all of them. And many are the same for all endsegments. Therefore the intersection of infinite endsegments is infinite.
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Natural numbers don't "leave", sets don't change.
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Call it as you like. Fact is that the function of endsegments is losing elements. The limit is the empty endsegment.
Your sequence of endsegments (which are each countably infinte) is indeed losing an element of N with each iteration. Losing an element is not the same as reducing an infinite set's size though.
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The size of the intersection remains infinite as long as all endsegments remain infinite (= as long as only infinite endsegments are considered).
Don't be so stupid. Endsegments are defined as infinite, all of them and each and every one of them. The intersection is empty.
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