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

Am Sat, 19 Oct 2024 20:04:36 +0200 schrieb WM:

On 19.10.2024 17:21, joes wrote:

Am Sat, 19 Oct 2024 15:38:53 +0200 schrieb WM:

On 19.10.2024 14:20, Alan Mackenzie wrote:

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

On 17.10.2024 23:22, 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.

What else would it be.

The interval occupied by the numbers is doubled when all numbers are
multiplied by 2. If even the second half, which has not been
multiplied,

You just said all numbers are multiplied. What is the "second half"?

What is the second half of N? What do we get when we double that.

Multiplying all n by 2 does not yield the same numbers.

In part it does. All even numbers are also naturals.

All even numbers which have been multiplied are natural numbers by
definition. And by the same definition there were not more even natural
numbers. But as the result there are more.

Look, if we construct the set of powers of 2, by starting with 1 and
multiplying (a finite number of times) by 2, this set is infinite
(and also bijective to N by "decorating" with 2^). Any number from that
set, when doubled, is also in that set.
Same with the even numbers: they can be divided into the disjunct sets
of the odd multiples of 2 and the multiples of 4. Every double of an
even number is a multiple of 4.
I'm sure you can generalise this.

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.