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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic
Subject: Re: Replacement of Cardinality
Date: Tue, 30 Jul 2024 21:28:48 -0400
Organization: i2pn2 (i2pn.org)
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On 7/30/24 1:37 PM, WM wrote:
> 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

And where is that in {1, 2, 3, ... w} ?

>>
>> 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.

Which is what number?

The input set was the Natural Numbers and w, so you are just proving you 
are lying,

>>
>>>
>>>> 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.

But you combined two different sets, so why can't there be a gap?

That is just the nature of unbounded sets.

> 
>>> 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.

Right, so we can say that ∀n ∈ ℕ: 1/n > 1/(n+1), so that for every unit 
fraction 1/n, there exists another unit fraction smaller than itself. 
The only way to have a smallest unit fraction is to have a largest 
natural number, at which point they aren't even "potentially infinite" 
as you have established a finite limit to them.

Remember, one property of Natural numbers that ∀n ∈ ℕ: n+1 exists.

> 
> Regards, WM