Re: Replacement of Cardinality

Liste des GroupesRevenir à s math 
Sujet : Re: Replacement of Cardinality
De : james.g.burns (at) *nospam* att.net (Jim Burns)
Groupes : sci.logic sci.math
Date : 19. Aug 2024, 17:58:10
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <c8abcebe-b826-45de-8507-71521bf2d6a7@att.net>
References : 1 2 3 4 5 6 7 8 9 10 11
User-Agent : Mozilla Thunderbird
On 8/19/2024 7:40 AM, WM wrote:
Le 17/08/2024 à 23:39, Jim Burns a écrit :
On 8/17/2024 9:28 AM, WM wrote:
Le 16/08/2024 à 19:39, Jim Burns a écrit :

no element of ℕᵈᵉᶠ is its upper.end,
because
for each diminishable k
diminishable k+1 disproves by counter.example
that k is the upper.end of ℕᵈᵉᶠ
>
SBZ(x) starts with 0 at 0 and increases,
but at no point x it increases by more than 1
because of
∀n ∈ ℕ: 1/n - 1/(n+1) > 0.
Therefore there is a smallest unit fractions
and vice versa a greatest natnumber.
What can't you understand?
>
How can ½⋅β
⎛ half the allegedly.positive greatest.lower.bound β of
⎝ visibleᵂᴹ.unit.fractions
be both lower.bound ( ½⋅β < β)
and not.lower.bound ( 2⋅β > ⅟k and ½⋅β > ¼⋅⅟k)
of the visibleᵂᴹ.unit.fractions?
>
Visible unit fractions have the lower bound 0.
Visibleᵂᴹ unit fractions have the greatest.lower.bound 0.
Otherwise, ½⋅β > 0 is contradictory.
⎛ not.bound 2⋅β > ⅟k
⎜ not.bound ½⋅β > ¼⋅⅟k
⎝ bound ½⋅β < β

Dark unit fractions have a smallest element.
Darkᵂᴹ unit.fractions are positive. Correct?
I don't want to alter your darkᵂᴹ

No smaller unit fractions is existing
Is that smallest darkᵂᴹ unit.fraction not a unit.fraction?
Is it not positive?
Each positive point,  thus,
  that smallest darkᵂᴹ unit.fraction
is not a lower.bound,  thus,
  has smaller unit.fractions, visibleᵂᴹ ones.
Are visibleᵂᴹ unit.fractions not existing?

No smaller unit fractions is existing
because no larger natnumber is existing.
For visibleᵂᴹ natural k
k∪{k} = k+1 disproves no larger.than.k existing.
Darkᵂᴹ natural 𝔊 has darkᵂᴹ unit.fraction ⅟𝔊
Darkᵂᴹ ⅟𝔊 is not.lower.bound of visiblesᵂᴹ ⅟k
Exists visibleᵂᴹ ⅟k < ⅟𝔊, thus k > 𝔊
For darkᵂᴹ natural 𝔊
k > 𝔊 disproves no larger.than.𝔊 existing.
Each natural, visibleᵂᴹ or darkᵂᴹ, is disproved
from no larger.than.it existing.
Each unit.fraction, visibleᵂᴹ or darkᵂᴹ, is disproved
from no smaller.than.it existing.

Date Sujet#  Auteur
27 Jul 24 * Re: Replacement of Cardinality902Jim Burns
28 Jul 24 +* Re: Replacement of Cardinality892WM
28 Jul 24 i`* Re: Replacement of Cardinality891Jim Burns
29 Jul 24 i +* Re: Replacement of Cardinality47Ross Finlayson
29 Jul 24 i i`* Re: Replacement of Cardinality46Ross Finlayson
29 Jul 24 i i `* Re: Replacement of Cardinality45Ross Finlayson
29 Jul 24 i i  `* Re: Replacement of Cardinality44Jim Burns
29 Jul 24 i i   `* Re: Replacement of Cardinality43Ross Finlayson
29 Jul 24 i i    +* Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity)14Ross Finlayson
1 Jan 25 i i    i`* Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity)13Ross Finlayson
1 Jan 25 i i    i `* Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity)12Jim Burns
2 Jan 25 i i    i  `* Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity)11Ross Finlayson
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2 Jan 25 i i    i   i`* Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity)3Jim Burns
2 Jan 25 i i    i   i `* Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity)2Ross Finlayson
2 Jan 25 i i    i   i  `- Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity)1Jim Burns
2 Jan 25 i i    i   `* Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity)6Jim Burns
2 Jan 25 i i    i    `* Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity)5Ross Finlayson
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3 Jan 25 i i    i     i`- Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity)1Ross Finlayson
2 Jan 25 i i    i     `* Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity)2Jim Burns
3 Jan 25 i i    i      `- Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity)1Ross Finlayson
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30 Jul 24 i i      `* Re: Replacement of Cardinality (ubiquitous ordinals)25Jim Burns
30 Jul 24 i i       `* Re: Replacement of Cardinality (ubiquitous ordinals)24Ross Finlayson
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4 Aug 24 i i          i   i       `* Re: Replacement of Cardinality (ubiquitous ordinals)5Jim Burns
4 Aug 24 i i          i   i        `* Re: Replacement of Cardinality (ubiquitous ordinals)4Ross Finlayson
4 Aug 24 i i          i   i         `* Re: Replacement of Cardinality (ubiquitous ordinals)3Jim Burns
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8 Aug 24 i i         i  i i    `* Re: Replacement of Cardinality (real-valued)4Ross Finlayson
8 Aug 24 i i         i  i i     `* Re: Replacement of Cardinality (real-valued)3Jim Burns
8 Aug 24 i i         i  i i      `* Re: Replacement of Cardinality (real-valued)2Ross Finlayson
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4 Aug 24 i i         i   `- Re: Replacement of Cardinality1WM
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