WM <
askas...@gmail.com> writes:
(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
Unendlichen" at Hochschule Augsburg.)
> Ben Bacarisse schrieb am Sonntag, 16. April 2023 um 23:30:38 UTC+2:
>> WM <
askas...@gmail.com> writes:
>> > Ben Bacarisse schrieb am Sonntag, 16. April 2023 um 02:29:34 UTC+2:
>>
>> >> If you want to reclaim some integrity, ask someone to prove that NUF is
>> >> constant in (0,1]
>> >
>> > That means ℵo unit fractions are between 0 and (0, 1].
>> No. NUF is provably constant in (0,1],
>
> Then there must be ℵ₀ unit fractions before the interval.
Are you seriously suggesting that there are unit fractions <= 0? Not
only does it seem like you are suggesting their existence, you are
suggesting that somehow they, magically, affect the cardinality of a set
they are not members of!
I think you are now just saying anything, no matter how daft.
The facts: NUF is a two-valued function, zero for x <= 0 and ℵ₀
otherwise. Functions with a discontinuity like this are nothing special
and sets so dense in the reals that the "number of them" in (0,x] is
infinite for all x > 0 are also common. All your arguments apply to
these sets as well. The positive rationals are all "at distinct points"
and have non-zero distances from each other yet NPR(x) = ℵ₀ for x > 0.
The positive irrationals are all "at distinct points" and have non-zero
distances from each other yet NPI(x) = c for x > 0...
> That means they cannot be discerned. They are dark.
They are not even unit fractions, so any description of them can be
taken to be vacuously true: the unit fractions that come before (0,1]
are all green. Yes! The reciprocals of the unit fractions that come
before (0,1] are all even primes. Yes!
>> That would be the paper of the century, and you'd win
>> medals an honours beyond imagining. Yet you prefer to post in this
>> forgotten corner of the Internet. I find that ... telling.
>
> The reason is simple. Matheologians don't want to be unmasked as
> charlatans.
Ah, you subscribe to the world-wide conspiracy theory. Is this why you
can only teach this stuff in an optional course in a college with no
mathematics program? I ask because, although I am a part of the
world-wide conspiracy, I don't always read the newsletters.
>> Mathematics
>> (what you call set theory) shows that NUF(x) is never zero in the
>> interval (0,1] so saying that it increases "from 0" in (0,1] is wrong.
>
> In increases in the interval [0, 1] from 0 to ℵ₀.
Yes (though the metaphor is a bad one).
> Since 0 is not unit fraction, it increases in the interval (0, 1] from
> 0 to ℵ₀.
No function can increase "from 0" in an interval if it never has the
value 0 in that interval! This is just a misuse of language. You find
the unbounded density of unit fractions near zero to have properties that
run counter to your intuition, but "how odd" does not constitute a proof.
The error above is simply one of words. There's no mathematical issue
here unless you claim that
∃x in (0,1] with NUF(x) = 0?
If so, please do say that. As you tell me: "use mathematics not words".
If you don't claim that, we are in agreement about the facts.
>> It is also wrong because NUF(x) is constant in (0,1] so saying that it
>> increases in (0,1] is absurd.
>
> If there are ℵ₀ unit fractions in the interval, then there are at
> least two unit fractions. They have a distance. Therefore one of them
> does not contribute to NUF(x) = ℵ₀ over the complete interval.
Use mathematics, not words:
Let UF(x) = { 1/n | n in N and 1/n < x }.
∀u1, u2 in UF(1) with u1 =/= u2
∃x in (0,1] with u1 in UF(x) and u2 not in UF(x)
or u2 in UF(x) and u1 not in UF(x)
> This arguments however holds for all unit fractions.
Word game alert: attempted exploitation of double meaning of "all"! Use
mathematics not words. The above holds for all pairs of unit fractions.
If you mean something else, write it property with the quantifiers in
the right order.
NUF(x) is constant in (0,1]. You know what you need to show to prove
otherwise, and you just can't do it.
--
Ben.