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

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

On 17.10.2024 20:39, Alan Mackenzie wrote:

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

There is a general rule not open to further discussion:
When doubling natural numbers we obtain even numbers which have not been
doubled.
In potential infinity we obtain more even natural numbers than have been
doubled.
In actual infinity we double ℕ and obtain neither ℕ or a subset of ℕ.

All of these "rules" are so loosely and ambiguously formulated, that they
don't actually say anything at all - they are meaningless.

These rules are basic.

They are not.  They're loosely worded and ambiguous.  They do not form
the basis of any further mathematics.

You don't understand them.  Perhaps too much at once.

Now we have an ad hominem.  I understand full well how meaningless they
are.

Start with 2n > n for every natural number.  (0 is not a natnumber.)

Depends on the exact formulation.  0 is frequently regarded as a natural
number.  It makes it easier to build, for example, rings on top of it.

But other than zero, 2n > n for every natural number, yes.  In
particular, for every natural number n, 2n is also a natural number.

If you can't understand or don't believe, then there is no common basis
for discussion.

It's not a matter of belief.  It's a matter of correct and rigorous
mathematics.

Regards WM

 

--
Alan Mackenzie (Nuremberg, Germany).