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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic
Subject: Re: Replacement of Cardinality
Date: Thu, 1 Aug 2024 19:53:02 -0400
Organization: i2pn2 (i2pn.org)
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On 8/1/24 8:27 AM, WM wrote:
> Le 01/08/2024 à 02:09, Richard Damon a écrit :
>> On 7/31/24 10:27 AM, WM wrote:
>>> 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.
>>
>> In other words, outside the Natural Nubmer, all of which are defined 
>> and definable.
> 
> That is simply nonsense. Do you know what an accumalation point is? 
> Every eps interval around 0 contains unit fractions which cannot be 
> separated from 0 by any eps. Therefore your claim is wrong.

And thus there is no "smallest" unit fraction, as for any eps, there are 
unit fractions smaller, and thus NUF(x) does have a value in the finites 
where it has the value of 1.

>>
>>>>
>>>> The input set was the Natural Numbers and w, 
>>>
>>> ω/10^10 and ω/10 are dark natural numbers.
>>
>> They may be "dark" but they are not Natural Numbers.
> 
> They are natural numbers.
>>
>> Natural numbers, by their definition, are reachable by a finite number 
>> of successor operations from 0.
> 
> That is the opinion of Peano and his disciples. It holds only for 
> potetial infinity, i.e., definable numbers.

No, it holds for ALL his numbers.

> 
>>> I assume completness.
>>
>> I guess you definition of "completeness" is incorrect.
>>
>> If I take the set of all cats, and the set of all doges, can there not 
>> be a gap between them?
> 
> What is the reason for the gap before omega? How large is it? Are these 
> questions a blasphemy?

Because it is between two different sorts of number.

There is a gap between 1 and 2, but that doesn't bother you.

>>
>>>
>>>>> ∀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.
>>
>> Right, for ALL n in ℕ, there exist another number in ℕ that is n+1,
> 
> That does ny formula not say. It says for all n which have successors, 
> there is  distance between 1/n and 1/(n+1).
>>>>
>>>> Remember, one property of Natural numbers that ∀n ∈ ℕ: n+1 exists.
>>>
>>> Not for all dark numbers.
>>
>> Maybe not for dark numbers, but it does for all Natural Numbers, as 
>> that is part of their DEFINITION.
> 
> It is the definition of definable numbers. Study the accumulation point. 
> Define (separate by an eps from 0) all unit fractions. Fail.

So, which Unit fraction doesn't have an eps that seperates it from 0?

You just get your order of conditions reversed.

For all 1/n, there is a eps that is smaller than it (like 1/(n+1) )

And for every eps, there is a unit fraction smaller than it

1 / (ceil (1/eps) + 1)

So we have an unlimited number of Unit fractions, and no smallest one.

> 
> Regards, WM