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

On 10/20/24 3:30 AM, WM 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. What else could twice mean?

>

.... containing numbers not in the original set ℕ.

>
No.  If you think that, then give an example of a 2n which "isn't in the
original set N".

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

You won't and you can't.  But you'll likely come back
to your standard get-out clause about (non existent) "dark numbers".

 The only alternative is that doubling numbers creates only doubled numbers. It is not acceptable. It cannot be apologized by the possibility to map all numbers on even numbers.
 Regards, WM
 

No, the alternative is that infinite sets work like infinite sets without an end, and that this means that some of the properties of finite sets don't hold for infinite sets.
The fact that this shows that all numbers can be mapped one-to-one to a proper subset of themselves, the even numbers, just is one of those properties.
If you can't handle it, you can't handle infinite sets.