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

On 11/06/2024 08:07 PM, Ross Finlayson wrote:

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.

>
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.
>
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 "<".
>
>

About "sum all of them", there's of course that the operations of
arithmetic and algebras modeling those are usually consider
"pair-wise", "dyadic", while the idea of those all associated
is "a higher order schema", with the orders of those each being
pair-wise then finite themselves. Then, the "univalent" is
basically the idea of "infinite union", about when it is so
that the universal quantifier has what's called transfer principle,
that what's so of the quantified/comprehended is what's so of
the quantified/combined. Then, in set theory at least, if
mostly though logic about a realm of objects called a universe
of set theory, the transfer principle and bridge results reflect
on things like "the limit is the sum", that otherwise most usually
of course is involved what's called "book-keeping", that the
structure must not lose its beans, in the great bean-counting,
the book-keeping.
Then, various approaches are considered to add univalency,
like homotopy type theory and some other considerations
what would arrive at what are perceived powerful when correct
schemes, yet, there must be the book-keeping or it's plainly
fraught with getting it all thoroughly wrong.
So, obviously somebody's already had the great idea of
adding up all the numbers before, for example Stevin
and as about the p-adic integers, infinite integers of sorts
that are distinct though. For most people though, such
is the milieu of "Big O". The idea of "Big O" is that
in the asymptotic, it results what's proportionate,
then as with regards to that in mathematics, that's
about what's called convergence. Then, you'd figure
that convergence would be thusly one of the most well-defined
things in mathematics, and it is and isn't. The theory
of complexity theory and chaos theory speak to, for example,
that after convergence, there's emergence, which is a sort
of higher-order convergence, after and through both the
pair-wise in the infinitary combinatorics, and the illative
(or univalent) when they don't exactly agree, and indeed
it's for law(s) of large numbers to help explain how the
transfer principle in logic makes the book-keeping, of
the Big O and the little o and the theta and various
parts of concrete mathematics and asymptotics.
Some of the Pythagoreans for example, given irrational
quantities like root two, would have given that "well
there are two infinite integers, their ratio is root two",
and maybe that's not Archimedean, but it isn't necessarily
not Pythagorean. There's a theory where it's so, and
all the finite numbers are the same.
Then, about Farey series or 1/n, and that going to zero,
then as with regards to the little-end of the geometric series,
it is so that both go to zero.