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Subject: Re: Replacement of Cardinality
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Date: Wed, 31 Jul 24 14:27:06 +0000
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From: WM <wolfgang.mueckenheim@tha.de>
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Le 31/07/2024 à 03:28, Richard Damon a écrit :
> 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} ?

In the midst, far beyond all definable numbers, far beyond ω/10^10.
> 
> The input set was the Natural Numbers and w, 

ω/10^10 and ω/10 are dark natural numbers. 

>>>> 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?

I assume completness.

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

No. My formula says ∀n ∈ ℕ.
> 
> Remember, one property of Natural numbers that ∀n ∈ ℕ: n+1 exists.

Not for all dark numbers.

Regards, WM