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From: joes <noreply@example.org>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Wed, 27 Nov 2024 18:02:15 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
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Am Wed, 27 Nov 2024 18:13:06 +0100 schrieb WM:
> On 27.11.2024 13:32, Richard Damon wrote:
>> On 11/27/24 5:12 AM, WM wrote:
> 
>>> Of course. |{1, 2, 3, 4, ...}| = |ℕ| and |{2, 3, 4, ...}| = |ℕ| - 1 is
>>> consistent.
>> 
>> So you think, but that is because you brain has been exploded by the
>> contradiction.
>> We can get to your second set two ways, and the set itself can't know
>> which.
>> We could have built the set by the operation of removing 1 like your
>> math implies, or we can get to it by the operation of increasing each
>> element by its successor, which must have the same number of elements,
> 
> Yes, the same number of elements, but not the same number of natural
> numbers.
> Decreasing every element in the real interval (0, 1] by one point
> yields the real interval [0, 1). The set of points remains the same, the
> set of positive points decreases by 1.
What is "decreasing by a point"?

> Replacing every element of the set {0, 1, 2, 3, ...} by its successor
> yields {1, 2, 3, ..., ω}. The number of ordinals remains the same, the
> number of finite ordinals decreases.
There is no natural whose successor is omega.

-- 
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.