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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Wed, 27 Nov 2024 12:58:15 -0500
Organization: i2pn2 (i2pn.org)
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On 11/27/24 12:13 PM, WM wrote:
> 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.
 >

Of course they are, if n is a Natural Number, Sn (S being the Successor 
operator) is also one.

> Hint: 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.

But what number changes "natural number" status?

> 
> 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.

Nope, because omega is NOT the successor for any natural number, the 
successor of EVERY Natural Number is a Natural Number.

Defintions you know.

It it the successor for the SET of natural numbers.

> 
> Regards, WM

So, you are just showing your ignorance of the definitions things you 
are talking about, because you are using a logic that has gone 
inconsistent and blown up your brain.