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

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:

When doubling natural numbers we obtain even numbers which have not
been doubled.
This is a sentence that every mathematician can understand.

It is not - it is ill formed and ambiguous.  It doesn't say which
natural numbers are being doubled.

That is not said because it is true for all sets of natural numbers.
Your critique is therefore not justified but due to your lack of
comprehension.

It is not true for infinite sets.

 It is unmathematical in that it
seems to posit a doubling being done one element at a time

Wrong again. Even all natural numbers can be multiplied by 2.

You seem unable to imagine that.

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.

"if"

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"?

contains natural numbers, then there are more after the
procedure than before. Hence the "set" has changed and therefore is not
a set. Note that sets do not change.

Tell that the potential infinity.

It is true because the interval covered by the doubled numbers is
twice as large as the interval covered by the numbers to be doubled.

The interval is infinite.  "Doubling" an infinite set yields a set of
the same size as the original - there is a 1-1 correspondence between
them.

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

In part it does. All even numbers are also naturals.
  {1, 2, 3, 4} n 2*{1, 2, 3, 4}
= {1, 2, 3, 4} n   {2, 4, 6, 8}
= {2, 4}


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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.