Re: how

On 6/16/2024 6:52 PM, Moebius wrote:

Am 16.06.2024 um 21:51 schrieb Moebius:

Am 16.06.2024 um 21:41 schrieb Moebius:

Concerning his claim
"But never two (different) unit fractions at
the same coordinate.":
it is so trivial that
it's hard to state/express it properly.
>
Unit fractions are real numbers.
And "two" unit fractions
(and hence real numbers) a, b are
either identical (a = b) or not (a =/= b). That's all.
>
Well, my last shot:
If a and b are two _different_ unit fractions, then there's no real number c
such that a = c = b
(though -of course- a and b are real numbers).

>
To put it in simple words:
/different/ means /not the same/.

We ask
what can we say (or not.say) about a and b
if they are (or are.not) the same?
In my opinion, that answer supersedes
a lot of interesting  philosophy about
being really the same.
Standardly, we can say a=a and b=b and,
whatever one can say about a, one can say about b
"What can we say?" squeezes our idea of "the same"
into a shape well.adapted for
finite not.first.false.only claim.sequences,
our method by which we explore infinity.
(I've beaten this drum before.)

So _different_ unit fractions
CAN'T BE "at the same coordinate"
(i.e. equal to the same real number).

Yes.
However, WM gives 'fraction' and 'rational'
different meanings.
1/2 2/4 3/6 4/8 ...
are all different _fractions_ corresponding to
the same _rational_ at the same point.
_As rationals_
whatever we can say about 1/2
we can say about 2/4
Conventionally. By definition.
_As fractions_
in Cantor's sequence,
the index of 1/2 is 2
the index of 2/4 is 12
We can say different things.