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

On 11/4/2024 6:07 AM, WM wrote:

On 03.11.2024 22:21, Richard Damon wrote:

But Induction *IS* valid for all Natural Numbercs.

>
Only for all natnumbers which
can be defined and which
belong to a finite initial segment which
is followed by ℵo natnumbers most of which you cannot define.

ℵ₀ is how many elements are in
the minimal set holding 0 and closed under n↦n+1
That is a set of only elements which
belong to a finite initial segment which
is followed by ℵ₀ elements each of which
belongs to a finite initial segment.
If you call something else ℕ
you aren't referring to that set,
which is what everyone else calls ℕ

By induction you can prove the sum n(n+1)/2
for every initial segment 1+2+3+...+n.

...every finite initial segment...

But not for all natural numbers.

No finite initial segment is all the natural numbers.
Anything countable.to is countable.past.
Anything countable.past is not.last.countable.to.
No countable.to is last.countable.to.