Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

On 11/07/2024 03:58 AM, Richard Damon wrote:

On 11/7/24 4:13 AM, WM wrote:

On 07.11.2024 03:27, Richard Damon wrote:

On 11/6/24 10:15 AM, WM wrote:

On 06.11.2024 12:46, Richard Damon wrote:

On 11/5/24 11:45 AM, WM wrote:

On 05.11.2024 04:08, Richard Damon wrote:

On 11/4/24 12:11 PM, WM wrote:

On 04.11.2024 13:14, Richard Damon wrote:

On 11/4/24 6:07 AM, WM wrote:

>

By induction you can prove the sum n(n+1)/2 for every initial
segment 1+2+3+...+n. But not for all natural numbers.

>

But all Natural Numbers can be defined.

>
All defined numbers can be summed. Not all natural numbers can
be summed.

>
Why not?

>
because most cannot be defined.

>
But they ARE defined.

>
Then sum all of them.

>

They will sum to Aleph_0

>
That is not a sum. It is just another name for infinity.

Note, Addition on Natural Numbers is closed for FINITE sums (the sum
of a finite number of numbers), not necessarily for infinite series
of them.

>
That is because infinitely many always contain dark numbers. All not
dark numbers can be summed.
>
Regards, WM
>

>
There are no dark numbers. You can't actually define what you mean by a
"dark number" except in ways that admit that you logic can't create the
actual infinite sets you want to talk about.
>
Sorry, logic based on the presumption of a lie is just broken.

How about there are none at all?
Infinity doesn't exist ...
Infinity - 1 doesn't exist, ...
Infinity - 2 doesn't exist, ...
...
Some have that logic based on any assumptions
as non-logical stipulations at all: is broken.
That's a bit inconvenient for usual and often
successful ad-hoc theories: only insofar as
saying they know-it-all.
Foundations, amigos, Foundations includes _all_ the numbers.
Then, about the little-end of the geometric series
or 1/2^n as n -> oo, is for symmetry, between great
and small.
Then, one might aver: "there's no great only
our small", then it's like, "well then I guess
you won't mind that the integers (-infinity, infinity)
is not a thing, you say", ....
Being contrarian, ..., instead of explanatory.