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

WM <wolfgang.mueckenheim@tha.de> wrote:

On 20.10.2024 11:39, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

On 19.10.2024 22:19, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

The result covers twice the interval, ....

It does not, except in the sense that twice infinite = infinite.

Twice the infinite contains numbers not in the first infinite.

It does not.

2n > n holds for all natural numbers.

Yes, it holds for all finite numbers greater than zero.  It does not hold
for infinite numbers.

What else could twice mean?

It doesn't really mean much at all.

It means that after doubling more numbers are there than have been doubled.

You could do with a better dictionary.

These numbers are dark. Their existence is proven by the fact that
∀n ∈ ℕ, n < 2n.

Their non-existence I proved in this thread, many posts ago.

Based on wrong assumptions.

No, based on standard mathematical knowledge.  You failed to challenge my
proof when I posted it, so to disparage it now is dishonest.

The only alternative is that doubling numbers creates only doubled
numbers.

That's more or less what happens, except that numbers aren't "created"..
They just are.

That's not what happens in mathematics.

How would you know?  You're not a mathematician.

It is not acceptable.

It is accepted by mathematicians.

Only by those who never thought about that topic.

It is accepted by ALL mathematicians, the ones who accept that 2 + 2 = 4.
It's pretty basic stuff, not at all advanced.

It cannot be apologized by the possibility to map all numbers on even
numbers.

There exists a 1-1 correspondence between all natural numbers and all
even natural numbers, a proper subset of all natural numbers.  The set of
all natural numbers is thus infinite.

The mapping needs larger numbers than have been mapped.

It does not.  Each number in the image is in the range.

That is not a matter of infinite sets but basic mathematics of all
natural numbers.  The correspondence between all natural numbers and
all even natural numbers requires a variable "all". Potential infinity.

It does not.  "All" means what it means.  "Potential" infinity is purely
a historical artifact with no currency in modern mathematical discourse.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).