Re: Replacement of Cardinality

On 8/11/2024 10:09 AM, WM wrote:

Le 10/08/2024 à 19:39, Jim Burns a écrit :

On 8/10/2024 11:59 AM, WM wrote:

Le 09/08/2024 à 05:34, Jim Burns a écrit :

and he thinks that
a set ordered with two ends is
more complete than
the same set with one or zero ends.

>
The set of unit fractions has two ends,
namely at 1 and before 0.

>
No.

>
Note:
And end is there, or before the point,
after which no elements follow.

⎛ Something not.in a set cannot be
⎜ an end of the set,
⎜ although it might be a bound of the set.
⎜
⎜ A greatest.lower.bound which is in the set
⎜ is its lower.end.
⎜
⎜ A greatest.lower.bound which isn't in the set
⎜ isn't its lower.end,  and
⎜ nothing else is its lower.end,
⎜ because
⎜ if it were its lower.end,
⎜ it would be its greatest.lower.bound,
⎜ which it isn't.
⎜
⎜⎛ If the greatest.lower.bound isn't an element,
⎝⎝ no lower.end exists.
⎛ The greatest.lower.bound of the unit.fractions
⎜ isn't a unit.fraction.
⎜
⎝ _No lower.end of the unit.fractions exists_
⎛ The greatest.lower.bound of the unit.fractions
⎜ isn't a unit.fraction.
⎜
⎜ Each unit.fraction is positive.
⎜ 0 is a lower.bound of the unit.fractions.
⎜
⎜( Each x > 0 is not a lower.bound of unit.fractions.
⎜
⎜ 0 is the greatest.lower.bound of unit.fractions.
⎜
⎝ 0 isn't a unit.fraction.
⎛ Each x > 0 is not a lower.bound of unit.fractions.
⎜
⎜⎛ Assume otherwise.
⎜⎜ Assume x > 0 is a lower.bound of unit.fractions.
⎜⎜
⎜⎜ Greatest.lower.bound β ≥ x > 0
⎜⎜
⎜⎜ Lower.bound ½⋅β < β
⎜⎜
⎜⎜ Not.lower.bound 2⋅β > β
⎜⎜ Unit.fraction ⅟k < 2⋅β
⎜⎜ Unit.fraction ¼⋅⅟k < ¼⋅2⋅β = ½⋅β
⎜⎜ Not.lower.bound ½⋅β
⎜⎝ Contradiction.
⎜
⎜ Therefore,
⎝ each x > 0 is not a lower.bound of unit.fractions.
⎛ Each x > 0 is not a lower.bound of unit.fractions.
⎜
⎜ 0 is the greatest.lower.bound of unit.fractions,
⎜ and 0 isn't a unit fraction.
⎜
⎝ _No lower.end of the unit.fractions exists_