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

On 11/06/2024 06:27 PM, 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.

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

>
Why not?

>
because most cannot be defined.

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But they ARE defined.

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Then sum all of them.
>
Regards, WM

>
They will sum to Aleph_0, since there is a countable infinite number of
them.
>
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.

It's probably usage to say "sums to infinity" or to say "Aleph_0-many"
not "sums to Aleph_0":  the cardinals have "exponentiation" yet
not "sums".
The only things defined on cardinals are "^" and "<".