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

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

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 not true for all sets of natural numbers.

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.

Which doesn't address the point I made.

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?  Here it
is your intuition which is wrong.  If you think you can obtain an
"undoubled" number in that mapping, please feel free to give an example.
You can't, of course, you'll just say that all such are "dark numbers",
your usual get out strategem, which is thoroughly predictable,
uninteresting, and false.

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.

You've got to be trolling here.  Please tell me you're trolling - nobody
can be that ignorant.

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.

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.

Nobody but you says it does.  It might be helpful if you actually
addressed the points I make, rather than using straw men.

Mathematics!

Mathematics is well worthy of study.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).