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

On 19.10.2024 16:24, Alan Mackenzie wrote:

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

rather than the standard mathematical concept of a mapping from N -> N
where n is mapped to 2n.  In this standard notion, all numbers are
doubled, and we encounter no undoubled even natural numbers.

 

Therefore the standard notion is wrong, if the natural numbers are a set.

 You mean it's wrong because it doesn't gel with your intuition?

No, it does not comply with mathematics. When multiplying all natural numbers by 2, then the number of numbers remains the same but the density is reduced and therefore the interval is doubled. 2 > n. Hence either natural numbers are created which have not been multiplied, then ℕ is not a set, or other numbers are created, then ℕ is a set.

If you think you can obtain an
"undoubled" number in that mapping, please feel free to give an example.

I can prove it by 2n > n.

You can't, of course, you'll just say that all such are "dark numbers",

Either dark numbers or natnumbers which have not been processed. There is no other way because 2n > n.

Note that I haven't talked about "sets which change" - that's entirely
your idea.  I talked about a map from N -> N, where n maps to 2n.

This Bourbaki-notion can be applied to potentially infinite sets only. Try to understand the correct mathematics.
Regards, WM