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

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

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.

How would you know?  You don't have a degree in maths, and aren't willing
to take lessons from those who do.

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.

That's not mathematics.  It's merely your intuition, derived from finite
sets and misapplied to infinite sets.

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.

Again, not mathematics, but merely your intuition.  We're talking about
infinite sets here and maps between them.  There is no notion of
"created" involved at all.

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.  You don't even understand the meaning of the word prove as it
pertains to mathematics.  As I say, if you maintain there is such a
doubled number which wasn't in N to begin with, you should produce it or
shut up.

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.

Again, not mathematics.  There is no notion of "processed" any more than
there's one of "created".  There's a map between two infinite sets.

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.

In mathematics, there is no meaningful distinction between what you think
of as two different forms of infinity.  That's why such a distinction has
fallen out of use in serious mathematical discourse.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).