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From: joes <noreply@example.org>
Newsgroups: sci.logic
Subject: Re: Replacement of Cardinality
Date: Fri, 2 Aug 2024 15:42:59 -0000 (UTC)
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Am Fri, 02 Aug 2024 15:09:18 +0000 schrieb WM:
> Le 02/08/2024 à 01:53, Richard Damon a écrit :
>> On 8/1/24 8:27 AM, WM wrote:
> 
>> And thus there is no "smallest" unit fraction, as for any eps, there
>> are unit fractions smaller,
> Your eps cannot be chosen small enough.
All unit fractions are larger than zero, so an epsilon can be chosen

>>> 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.
> Not for ℵo, i.e., for most it is wrong:
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> 
>>> 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 no gap above zero but e real continuum.
>
>> There is a gap between 1 and 2, but that doesn't bother you.
> All gaps of size 1 do not bother me..
> 
>>> 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?
> There are infinitely many by the definition of accumulation point. You
> cannot find them. Therefore they are dark.
That is not the definition. The "infinitely many" are not the same ones
for every epsilon. You can't seem to imagine different infinities.

>> You just get your order of conditions reversed.
> I get it the only corect way. Every eps that you can chose belongs to a
> set of chosen eps. This set has a minimum - at every time. It is finite.
> Quantifiers therefore can be reversed.
The set of reals is infinite and does not have a minimum. 

>> For all 1/n, there is a eps that is smaller than it (like 1/(n+1) )
> For all 1/n that you can define.
>
>> And for every eps, there is a unit fraction smaller than it
> There are infinitely many, namely almost all.
>
>> So we have an unlimited number of Unit fractions, and no smallest one.
> But you have a limited number of eps.
WTF?

-- 
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.