Re: Replacement of Cardinality (infinite middle)

On 8/8/2024 8:26 PM, Ross Finlayson wrote:

On 08/08/2024 03:30 AM, FromTheRafters wrote:

on 8/8/2024, WM supposed :

Le 08/08/2024 à 00:17, Moebius a écrit :

Actually, his "thinking process" is simple:
"Since there is a gap (space) between
adjacent unit fractions and
all unit fractions are in the interval (0, 1],
there must be FINITELY MANY of them
(i.e. a first/smallest one)."

>
No, that is nonsense.
There are not finitely many unit fractions.

>
Then stop assuming that
there is a first and last element.

>
Of course, you can start with a first and last element,
then make infinitely-many in the middle.
>
0 ... ( ... infinitely-many ... ) ... infinity

⎛ A finite order is trichotomous and
⎜ each non.{} subset is two.ended.
⎜
⎜ An infinite order is trichotomous and
⎜ at least one subset is one. or zero.ended
⎜
⎝ No set has both a finite and an infinite order.
An infinite set might or might not have a first element.
What an infinite set MUST have is SOME non.{} SUBSET
without a first or without a last,
but that subset need not be the whole set.
A set which MUST have a first element in any order
is a finite set.
WM doesn't seem to know what he agrees to
when he agrees that
there are infinitely.many unit.fractions
and then argues as though
there are finitely.many unit.fractions
(because logicᵂᴹ).