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Subject: Re: Replacement of Cardinality
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Date: Tue, 30 Jul 24 17:37:09 +0000
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From: WM <wolfgang.mueckenheim@tha.de>
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Le 30/07/2024 à 03:18, Richard Damon a écrit :
> On 7/29/24 9:11 AM, WM wrote:

>>> But what number became ω when doubled?

 ω/2
> 
> No, that is w double, what number in the first set became w?
> 
>>>
>>> Every natural number when doubled is a Natural Number.
>> 
>> No.
> 
> Why not? WHich ones don't?

 ω/2 and larger.
> 
>> 
>>> Note, ω-1 doesn't exist in the base transfinite numbers, just as -1 
>>> doesn't exist in the Natural Numbers, you can't go below the first 
>>> element.
>> 
>> If all natural numbers exist, then ω-1 exists.
> 
> Why?

Because otherwise there was a gap below ω.

>> That is unavoidable. You believe in the magical appearance of infinitely 
>> many unit fractions. That breaks logic and mathematics.
> 
> Nope,

∀n ∈ ℕ: 1/n - 1/(n+1) > 0. Note the u niversal quantifier.

Regards, WM