On 8/25/2021 3:42 PM, WM wrote:
> William schrieb am Mittwoch, 25. August 2021 um 14:54:01 UTC+2:
>> On Wednesday, August 25, 2021 at 9:26:53 AM UTC-3, WM wrote:
>>> William schrieb am Mittwoch, 25. August 2021 um 01:56:46 UTC+2:
>>>> ... there is no rational that you cannot write down that is not mapped to.
>>>
>>> What about the 99.9 % missing in interval (1000, 1001]?
>> Percentages applied to infinite cardinalities give nonsense.
>
> Percetage applied to density gives mathematical result: https://mathoverflow.net/questions/362791/what-fraction-of-fractions-does-cantors-famous-sequence-enumerate

wrong. that is a question, NOT a result.

>
>> There is no rational in (1000,1001] that is not mapped to.
>
> But there are less mapped in (1000,1001] than in (0,1]. That is not doubted. What went wrong?

wrong. All rationals are mapped.

IF you have a issue, use MATH

>
>>> What about the first unit fraction when starting from 0?
>>>
>> This does not exist.
>
> What about the first that exists?

which one ? can you name it ?

>
> Regards, WM
>

On 8/25/2021 3:39 PM, WM wrote:
> William schrieb am Mittwoch, 25. August 2021 um 14:38:50 UTC+2:

<snip troll bait>

onsdag 25 augusti 2021 kl. 14:18:15 UTC+2 skrev WM:
> Greg Cunt schrieb am Dienstag, 24. August 2021 um 23:20:11 UTC+2:
> > On Tuesday, August 24, 2021 at 10:43:27 PM UTC+2, WM wrote:
> >
> > > There are the first thre steps: 1, 2, 3. There are even more. If they cease then <bla bla>
> >
> > THE DON'T CEASE,
> >
> > There is no largest natural number.
> Simple minds may think so. But when considering the unit fractions, then you should be able to see that somewhe zero is reached. Then the smallest unit fraction has been passed.
>
> Regards, WM

no, anyone know there is no largest natural numbers because no matter what you say is largest, I add 1 and tada, it is even larger!

There is no smallest unit fraction, there is no last unit fraction and if all you do is go through unit fractions, you NEVER get 0!

onsdag 25 augusti 2021 kl. 14:29:18 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Mittwoch, 25. August 2021 um 07:07:51 UTC+2:
> > tisdag 24 augusti 2021 kl. 22:57:38 UTC+2 skrev WM:
>
> > ALL rational are indiced. It is trivial to show there is a bijection betwee N and Q+
> It is trivial to show the contrary:
> ∀ q ∈ (0, 1] ∃ q+n ∈ (n, n+1] und ∀ q+n ∈ (n, n+1] ∃ q ∈ (0, 1]
> 1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, ...
> Regards, WM

That does not go against my statement that there is a bijection between N and Q+

zelos...@gmail.com schrieb am Donnerstag, 26. August 2021 um 07:15:08 UTC+2:
> onsdag 25 augusti 2021 kl. 14:29:18 UTC+2 skrev WM:

> > > ALL rational are indiced. It is trivial to show there is a bijection betwee N and Q+
> > It is trivial to show the contrary:
> > ∀ q ∈ (0, 1] ∃ q+n ∈ (n, n+1] und ∀ q+n ∈ (n, n+1] ∃ q ∈ (0, 1]
> > 1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, ...

> That does not go against my statement that there is a bijection between N and Q+

Not against a bijection between the fractions enumerated in (0, 1] and (1000, 1001]. But it goes against the assumption, that Cantor enumerates all fractions. There are less enumerated in (1000, 1001] than in (0, 1] but there are not less in (1000, 1001]. This excludes a bijection between |N and Q+ by lack of surjectivity.

Regards, WM

William schrieb am Donnerstag, 26. August 2021 um 01:20:41 UTC+2:
> On Wednesday, August 25, 2021 at 5:42:20 PM UTC-3, WM wrote:
> > But there are less mapped in (1000,1001] than in (0,1].
> Nope; there are aleph_0 mapped in (0,1] and aleph_0 mapped in (1000,1001]..
>
Yes, there is a bijection between the fractions enumerated in (0, 1] and in (1000, 1001]. But Cantor does not enumerates all fractions. There are less enumerated in (1000, 1001] than in (0, 1] but there are not less in (1000, 1001]. This excludes a bijection between |N and Q+ by lack of surjectivity..

1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, ... contains half of all fractions in the first unit interval. It is clearer here https://en.wikipedia.org/wiki/File:Diagonal_argument.svg

But there are more in every other interval than are enumerated there. This can even proved by Cantor's method. Here it is proved for the second unit interval:
1/1 + 1, 1/2 + 1, 2/1 + 1, 1/3 + 1, 3/1 + 1, 1/4, ...
Now half of all are in the second unit interval. Why is this so? Dark numbers are the reason.

Regards, WM

William schrieb am Donnerstag, 26. August 2021 um 01:12:57 UTC+2:

> "Number of points" is nonsense when you try to apply it to sets with infinite cardinality.

Cardinality is nonsense. Mathematics of infinite sequences shows that in https://en.wikipedia.org/wiki/File:Diagonal_argument.svg there are as many entries below the diagonal as above the diagonal.

Regards, WM

Jim Burns schrieb am Mittwoch, 25. August 2021 um 23:51:47 UTC+2:

> Cantor succeeds.

There is a bijection between the fractions enumerated in (0, 1] and in (1000, 1001]. But Cantor does not enumerates all fractions. There are less enumerated in (1000, 1001] than in (0, 1] but there are not less in (1000, 1001]. This excludes a bijection between |N and Q+ by lack of surjectivity.

1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, ... contains half of all fractions in the first unit interval. It is clearer here https://en.wikipedia.org/wiki/File:Diagonal_argument.svg

But there are more in every other interval than are enumerated there. This can even proved by Cantor's method. Here it is proved for the second unit interval:
1/1 + 1, 1/2 + 1, 2/1 + 1, 1/3 + 1, 3/1 + 1, 1/4, ...
Now half of all are in the second unit interval. Why is this so? Dark numbers are the reason.

Regards, WM

Greg Cunt schrieb am Mittwoch, 25. August 2021 um 23:42:58 UTC+2:
> On Wednesday, August 25, 2021 at 10:42:20 PM UTC+2, WM wrote:
> > William schrieb am Mittwoch, 25. August 2021 um 14:54:01 UTC+2:
> > > On Wednesday, August 25, 2021 at 9:26:53 AM UTC-3, WM wrote:
> > > >
> > > > What about the first unit fraction when starting from 0?
> > > >
> > > This does not exist.
> > >
> > What about the first that exists?
> Holy shit!

Does none exist?

Regards, WM

Jim Burns schrieb am Mittwoch, 25. August 2021 um 20:30:42 UTC+2:

> How do you count the sheep in a pasture?
>
The rational numbers in the intervals (n, n+1] are easier to compare.

∀ q ∈ (0, 1] ∃ q+n ∈ (n, n+1] and ∀ q+n ∈ (n, n+1] ∃ q ∈ (0, 1]

> We cannot count through the natural numbers until
> there are no more natural numbers.

We can by using mathematics of infinite sequences prove that half of all enumerated rationals are in the first interval. We can prove that there are at least as many rationals in the second interval. But less are enumerated there. This shows that surjectivity fails.

> That's wrong. What's wrong about it is the use of
> "half the elements" when we do not count until
> there are no more.

Half is easily seen by the matrix https://en.wikipedia.org/wiki/File:Diagonal_argument.svg

Regards, WM

William schrieb am Mittwoch, 25. August 2021 um 14:54:01 UTC+2:
> On Wednesday, August 25, 2021 at 9:26:53 AM UTC-3, WM wrote:
> > William schrieb am Mittwoch, 25. August 2021 um 01:56:46 UTC+2:
> > > ... there is no rational that you cannot write down that is not mapped to.
> >
> > What about the 99.9 % missing in interval (1000, 1001]?
> Percentages applied to infinite cardinalities give nonsense.

Yes, cardinalities are nonsense.

> There is no rational in (1000,1001] that is not mapped to.

No definable rational is missing. But 99.9 % less are enumerated.

> > What about the first unit fraction when starting from 0?
> >
> This does not exist.
>
What about the first existing unit fraction? Or are there no unit fractions existing?

Regards, WM

On 8/26/2021 7:54 AM, WM wrote:
> Greg Cunt schrieb am Mittwoch, 25. August 2021 um 23:42:58 UTC+2:
>> On Wednesday, August 25, 2021 at 10:42:20 PM UTC+2, WM wrote:
>>> William schrieb am Mittwoch, 25. August 2021 um 14:54:01 UTC+2:
>>>> On Wednesday, August 25, 2021 at 9:26:53 AM UTC-3, WM wrote:
>>>>>
>>>>> What about the first unit fraction when starting from 0?
>>>>>
>>>> This does not exist.
>>>>
>>> What about the first that exists?
>> Holy shit!
>
> Does none exist?
>
> Regards, WM
>

none exist.

proof;

1. assume k is the first unit fraction in [0,1]

2. however k/2 is closer to 0, therfore now first

3. Assumption is proven false.

4 there is no number that is "first" unit fraction.

On 8/26/2021 7:46 AM, WM wrote:
> William schrieb am Donnerstag, 26. August 2021 um 01:20:41 UTC+2:
>> On Wednesday, August 25, 2021 at 5:42:20 PM UTC-3, WM wrote:
>
>>> But there are less mapped in (1000,1001] than in (0,1].
>> Nope; there are aleph_0 mapped in (0,1] and aleph_0 mapped in (1000,1001].
>>
> Yes, there is a bijection between the fractions enumerated in (0, 1] and in (1000, 1001].

wrong, show your work.

> But Cantor does not enumerates all fractions.

wrong.

>There are less enumerated in (1000, 1001] than in (0, 1]

wrong.

> but there are not less in (1000, 1001]. This excludes a bijection between |N and Q+ by lack of surjectivity.

wrong.

>
> 1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, ... contains half of all fractions in the first unit interval. It is clearer here https://en.wikipedia.org/wiki/File:Diagonal_argument.svg

wrong

>
> But there are more in every other interval than are enumerated there.

wrong.

This can even proved by Cantor's method. Here it is proved for the second unit interval:
> 1/1 + 1, 1/2 + 1, 2/1 + 1, 1/3 + 1, 3/1 + 1, 1/4, ...

wrong. you have not shown that all fraction in second unit interval are there.

> Now half of all are in the second unit interval.

Wrong. you apply finite methods to infinite, Fail.

> Why is this so?

Wrong.

> Dark numbers are the reason.

wrong, Dark numbers are your excuse for not using Math, Fail.

>
> Regards, WM
>

TEN wrong is one post, *WM is always wrong*.

On 8/26/2021 8:02 AM, WM wrote:
> William schrieb am Mittwoch, 25. August 2021 um 14:54:01 UTC+2:
>> On Wednesday, August 25, 2021 at 9:26:53 AM UTC-3, WM wrote:
>>> William schrieb am Mittwoch, 25. August 2021 um 01:56:46 UTC+2:
>>>> ... there is no rational that you cannot write down that is not mapped to.
>>>
>>> What about the 99.9 % missing in interval (1000, 1001]?
>> Percentages applied to infinite cardinalities give nonsense.
>
> Yes, cardinalities are nonsense.

wrong.

>
>> There is no rational in (1000,1001] that is not mapped to.
>
> No definable rational is missing.

wrong, your "definable" is not a math term

> But 99.9 % less are enumerated.

wrong. all rationals are enumerated.

>
>>> What about the first unit fraction when starting from 0?
>>>
>> This does not exist.
>>
> What about the first existing unit fraction?

Wrong, there is no first

> Or are there no unit fractions existing?

lost you are, unit fractions exist

>
> Regards, WM
>

four wrongs and one lost. Every sentence in this WM post is wrong

On 8/26/2021 7:59 AM, WM wrote:
> Jim Burns schrieb am Mittwoch, 25. August 2021 um 20:30:42 UTC+2:
>
>
>> How do you count the sheep in a pasture?
>>
> The rational numbers in the intervals (n, n+1] are easier to compare.
>
> ∀ q ∈ (0, 1] ∃ q+n ∈ (n, n+1] and ∀ q+n ∈ (n, n+1] ∃ q ∈ (0, 1]

wrong

>
>> We cannot count through the natural numbers until
>> there are no more natural numbers.
>
> We can by using mathematics of infinite sequences prove that half of all enumerated rationals are in the first interval.

wrong.

> We can prove that there are at least as many rationals in the second interval.

wrong
>
>But less are enumerated there.

wrong

>This shows that surjectivity fails.

wrong.
>
>> That's wrong. What's wrong about it is the use of
>> "half the elements" when we do not count until
>> there are no more.
>
> Half is easily seen by the matrix https://en.wikipedia.org/wiki/File:Diagonal_argument.svg

wrong again

>
> Regards, WM
>

six wrongs in one post, every sentence again, its WM the troll fishing...

On Thursday, August 26, 2021 at 9:46:43 AM UTC-3, WM wrote:
> William schrieb am Donnerstag, 26. August 2021 um 01:20:41 UTC+2:
> > On Wednesday, August 25, 2021 at 5:42:20 PM UTC-3, WM wrote:
>
> > > But there are less mapped in (1000,1001] than in (0,1].
> > Nope; there are aleph_0 mapped in (0,1] and aleph_0 mapped in (1000,1001].
> >
> Yes, there is a bijection between the [rationals] enumerated in (0, 1] and in (1000, 1001].

Indeed, there is a bijection, that is a mapping that is both injective and surjective.

--
William Hughes

On Thursday, August 26, 2021 at 9:50:32 AM UTC-3, WM wrote:
> William schrieb am Donnerstag, 26. August 2021 um 01:12:57 UTC+2:
>
> > "Number of points" is nonsense when you try to apply it to sets with infinite cardinality.
> Cardinality is nonsense.

Nope, As you note there is a bijection between the rationals in (0,1] and the rationals in (1000,1001]. So the two sets have the same cardinality. It is nonsense to say that the range of the bijection has "more points] in (0,1] than in (1000,1001].

--
William Hughes

On Thursday, August 26, 2021 at 10:02:15 AM UTC-3, WM wrote:
> William schrieb am Mittwoch, 25. August 2021 um 14:54:01 UTC+2:
s

> > > What about the first unit fraction when starting from 0?
> > >
> > This does not exist.
> >
> What about the first existing unit fraction? Or are there no unit fractions existing?

There are an infinite number of existing elements of U. However, no element of U has the property that is it first.

--
William Hughes

William schrieb am Donnerstag, 26. August 2021 um 17:34:12 UTC+2:
> On Thursday, August 26, 2021 at 10:02:15 AM UTC-3, WM wrote:
> > William schrieb am Mittwoch, 25. August 2021 um 14:54:01 UTC+2:
> s
> > > > What about the first unit fraction when starting from 0?
> > > >
> > > This does not exist.
> > >
> > What about the first existing unit fraction? Or are there no unit fractions existing?
> There are an infinite number of existing elements of U. However, no element of U has the property that is it first.

How is that possible? Some unit fraction must be next to zero. Without dark unit fractions there would be a first one, next to zero. Only if it is dark, "next to" has no meaning.

Regards, WM

On Thursday, August 26, 2021 at 3:12:41 PM UTC-3, WM wrote:
> William schrieb am Donnerstag, 26. August 2021 um 17:34:12 UTC+2:
> > On Thursday, August 26, 2021 at 10:02:15 AM UTC-3, WM wrote:
> > > William schrieb am Mittwoch, 25. August 2021 um 14:54:01 UTC+2:
> > s
> > > > > What about the first unit fraction when starting from 0?
> > > > >
> > > > This does not exist.
> > > >
> > > What about the first existing unit fraction? Or are there no unit fractions existing?
> > There are an infinite number of existing elements of U. However, no element of U has the property that is it first.
> How is that possible?

Simple consequence of the fact that no element of the set |N_F has the property that it is last.

--
William Hughes

William schrieb am Donnerstag, 26. August 2021 um 17:26:38 UTC+2:
> On Thursday, August 26, 2021 at 9:50:32 AM UTC-3, WM wrote:
> > William schrieb am Donnerstag, 26. August 2021 um 01:12:57 UTC+2:
> >
> > > "Number of points" is nonsense when you try to apply it to sets with infinite cardinality.
> > Cardinality is nonsense.
> Nope, As you note there is a bijection between the rationals in (0,1] and the rationals in (1000,1001]. So the two sets have the same cardinality. It is nonsense to say that the range of the bijection has "more points] in (0,1] than in (1000,1001].

My bijection f(q) = q + 1000 shows same number of rational points because I cover all rationals including the dark ones. Same likr f(q) = q.

For the definable points however we find that half of all indexed rational points are in (0, 1]:

1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, ...

Less are indexed in (1000, 1001]. So many are missing there. But Cantor could also include them in his mapping by devising the alternative sequence

1/1 + 1000, 1/2 + 1000, 2/1 + 1000, 1/3 + 1000, 3/1 + 1000, 1/4 + 1000, ...

You see, now half of all indexed rational points are in (1000, 1001]. Rather arbitrary, this indexing.

Regards, WM

William schrieb am Donnerstag, 26. August 2021 um 20:18:53 UTC+2:
> On Thursday, August 26, 2021 at 3:12:41 PM UTC-3, WM wrote:
> > William schrieb am Donnerstag, 26. August 2021 um 17:34:12 UTC+2:
> > > On Thursday, August 26, 2021 at 10:02:15 AM UTC-3, WM wrote:
> > > > William schrieb am Mittwoch, 25. August 2021 um 14:54:01 UTC+2:
> > > s
> > > > > > What about the first unit fraction when starting from 0?
> > > > > >
> > > > > This does not exist.
> > > > >
> > > > What about the first existing unit fraction? Or are there no unit fractions existing?
> > > There are an infinite number of existing elements of U. However, no element of U has the property that is it first.
> > How is that possible?
> Simple consequence of the fact that no element of the set |N_F has the property that it is last.

For the definable elements this is true. There are infinitely many following upon every defined element. But if they do not reach to zero, then there is a gap.

Regards, WM

William schrieb am Donnerstag, 26. August 2021 um 17:17:13 UTC+2:
> On Thursday, August 26, 2021 at 9:46:43 AM UTC-3, WM wrote:
> > William schrieb am Donnerstag, 26. August 2021 um 01:20:41 UTC+2:
> > > On Wednesday, August 25, 2021 at 5:42:20 PM UTC-3, WM wrote:
> >
> > > > But there are less mapped in (1000,1001] than in (0,1].
> > > Nope; there are aleph_0 mapped in (0,1] and aleph_0 mapped in (1000,1001].
> > >
> > Yes, there is a bijection between the [rationals] enumerated in (0, 1] and in (1000, 1001].
>
> Indeed, there is a bijection, that is a mapping that is both injective and surjective.

Alas it is not between all fractions of these intervals. As Cantor shows us there are more hits in the first interval.

Regards, WM

On 8/26/2021 1:12 PM, WM wrote:
> William schrieb am Donnerstag, 26. August 2021 um 17:34:12 UTC+2:
>> On Thursday, August 26, 2021 at 10:02:15 AM UTC-3, WM wrote:
>>> William schrieb am Mittwoch, 25. August 2021 um 14:54:01 UTC+2:
>> s
>>>>> What about the first unit fraction when starting from 0?
>>>>>
>>>> This does not exist.
>>>>
>>> What about the first existing unit fraction? Or are there no unit fractions existing?
>> There are an infinite number of existing elements of U. However, no element of U has the property that is it first.
>
> How is that possible? Some unit fraction must be "next to" zero.

no, no number exists with that property

Proof;

1. assume k is the first unit fraction in [0,1]

2. however k/2 is closer to 0, therefore k/2 is now first

3. therefore, the Assumption is false.

4. there is no number that is "first" unit fraction.

On 8/26/2021 1:23 PM, WM wrote:
> William schrieb am Donnerstag, 26. August 2021 um 20:18:53 UTC+2:
>> On Thursday, August 26, 2021 at 3:12:41 PM UTC-3, WM wrote:
>>> William schrieb am Donnerstag, 26. August 2021 um 17:34:12 UTC+2:
>>>> On Thursday, August 26, 2021 at 10:02:15 AM UTC-3, WM wrote:
>>>>> William schrieb am Mittwoch, 25. August 2021 um 14:54:01 UTC+2:
>>>> s
>>>>>>> What about the first unit fraction when starting from 0?
>>>>>>>
>>>>>> This does not exist.
>>>>>>
>>>>> What about the first existing unit fraction? Or are there no unit fractions existing?
>>>> There are an infinite number of existing elements of U. However, no element of U has the property that is it first.
>>> How is that possible?
>> Simple consequence of the fact that no element of the set |N_F has the property that it is last.
>
> For the definable elements this is true. There are infinitely many following upon every defined element. But if they do not reach to zero, then there is a gap.
>
> Regards, WM
>

your mind has already been there,

80 Ants in an Interval
81 Ants in a Gap

476 Gap Ants
477 Gap Fearing Ants

338 definable Ants
339 definable natnumbers Ants
340 definable Ants in the cover
341 Definable Endsegment Ants
342 Definable but Not Defined Yet Ants
343 Definable and Dark Ants
344 Definable STFU Ants
345 Definable Ants that are not definable
346 Definable Ants vs Not Definable Ants
347 Define Undefinable Ants
348 Defined Ants
349 Defined Ants that Rap, Tap, Flash, Buzz and Hoof
350 degenerate Ants
351 Delusional Ants
352 Delusional Ants getting worse by the Day