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

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:

 

If you can't understand or don't believe, then there is no common basis
for discussion.

 

It's not a matter of belief.  It's a matter of correct and rigorous
mathematics.

 

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 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.

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.
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, contains natural numbers 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.

 

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.
Mathematics!
Regards, WM