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Am Tue, 08 Oct 2024 12:40:26 +0200 schrieb WM:If it exists then this point is next to zero.On 08.10.2024 12:04, Alan Mackenzie wrote:WM <wolfgang.mueckenheim@tha.de> wrote:But not the point inbetween?Points either are or are not. The points that are include one point nextHence all must be visible including the point next to zero, but theyThere is no point next to zero.
are not.
to zero.
All of them differ by a finite set of numbers (whoich is irrelevant) but contain an infinite set of numbers in common.The infinite sets contain what? No natural numbers? Natural numbersThese are infinite sets: {2, 3, 4, …}, {3, 4, 5, …}, {4, 5, 6, …}.
dancing around, sometimes being in a set, sometimes not? An empty
intersection requires that the infinite sets have different elements.
They contain all naturals larger than a given one, and nothing else.
Every natural is part of a finite number of these sets (namely, its
own value is that number). The set {n+1, n+2, …} does not contain n
and is still infinite; there are (trivially) infinitely many further
such sets. All of them differ.
If every single set is infinite, then the intersection is infinite too. These sets have lost some natural numbers but have kept infinitely many.Shrinking sets which remain infinite have not lost all elements.This goes for every single of these sets, but not for their infinite(!)
intersection.
If you imagine this as potential infinity,No, in potential infinity there are no endsegments.
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