William schrieb am Freitag, 20. August 2021 um 17:28:19 UTC+2:
> On Friday, August 20, 2021 at 8:09:53 AM UTC-4, WM wrote:
> > William schrieb am Donnerstag, 19. August 2021 um 22:48:48 UTC+2:
> > > On Thursday, August 19, 2021 at 2:35:52 PM UTC-4, WM wrote:
> > > > William schrieb am Donnerstag, 19. August 2021 um 20:18:13 UTC+2:
> > > > > On Thursday, August 19, 2021 at 1:53:53 PM UTC-4, WM wrote:
> > > > > > William schrieb am Donnerstag, 19. August 2021 um 19:24:31 UTC+2:
> > > > > > >
> > > > > you cannot use a step by step method (something you call a "well ordered way") to talk about the "completetion" of a set with no last element.
> > > > > > You cannot use it at all for infinite sets.
> > > > > Use other methods. Get results.
> > > > The methods you use are based upon bijections,
> > > Nope, Non step by step methods lead to bijections.
> > For every [element of the set of FISONs ] a bijection with |N can be analyzed step by step
>
> Look! Over there! A Pink Elephant!
> The bijection between |N and the *set of FISONs* can be analyzed step by step.

Only as long as the FISONs last. Then always aleph_0 natnumbers remain not yet covered.

Regards, WM

On 8/20/2021 4:36 PM, WM wrote:
> William schrieb am Freitag, 20. August 2021 um 17:28:19 UTC+2:
>> On Friday, August 20, 2021 at 8:09:53 AM UTC-4, WM wrote:
>>> William schrieb am Donnerstag, 19. August 2021 um 22:48:48 UTC+2:
>>>> On Thursday, August 19, 2021 at 2:35:52 PM UTC-4, WM wrote:
>>>>> William schrieb am Donnerstag, 19. August 2021 um 20:18:13 UTC+2:
>>>>>> On Thursday, August 19, 2021 at 1:53:53 PM UTC-4, WM wrote:
>>>>>>> William schrieb am Donnerstag, 19. August 2021 um 19:24:31 UTC+2:
>>>>>>>>
>>>>>> you cannot use a step by step method (something you call a "well ordered way") to talk about the "completetion" of a set with no last element.
>>>>>>> You cannot use it at all for infinite sets.
>>>>>> Use other methods. Get results.
>>>>> The methods you use are based upon bijections,
>>>> Nope, Non step by step methods lead to bijections.
>>> For every [element of the set of FISONs ] a bijection with |N can be analyzed step by step
>>
>> Look! Over there! A Pink Elephant!
>> The bijection between |N and the *set of FISONs* can be analyzed step by step.
>
> Only as long as the FISONs last. Then always aleph_0 natnumbers remain not yet covered.
>
> Regards, WM
>

yes, the old stop at k trick. and after k there is aleph_0 natnumbers... (duh)

how many years have you been chewing this cud ?

On Friday, August 20, 2021 at 5:36:44 PM UTC-4, WM wrote:
> William schrieb am Freitag, 20. August 2021 um 17:28:19 UTC+2:
> > On Friday, August 20, 2021 at 8:09:53 AM UTC-4, WM wrote:
> > > William schrieb am Donnerstag, 19. August 2021 um 22:48:48 UTC+2:
> > > > On Thursday, August 19, 2021 at 2:35:52 PM UTC-4, WM wrote:
> > > > > William schrieb am Donnerstag, 19. August 2021 um 20:18:13 UTC+2:
> > > > > > On Thursday, August 19, 2021 at 1:53:53 PM UTC-4, WM wrote:
> > > > > > > William schrieb am Donnerstag, 19. August 2021 um 19:24:31 UTC+2:
> > > > > > > >
> > > > > > you cannot use a step by step method (something you call a "well ordered way") to talk about the "completetion" of a set with no last element.
> > > > > > > You cannot use it at all for infinite sets.
> > > > > > Use other methods. Get results.
> > > > > The methods you use are based upon bijections,
> > > > Nope, Non step by step methods lead to bijections.
> > > For every [element of the set of FISONs ] a bijection with |N can be analyzed step by step
> >

So what? Giving a property of every *element of set tof FISONs* tells us nothing about whether the *set of FISONs* has this property.
It does not. The *set of FISONs* cannot be analyzed step by step. The *set of FISONs* can be analyzed if you use the appropriate tools.

--
William Hughes

On 8/20/2021 5:36 PM, WM wrote:
> William schrieb
> am Freitag, 20. August 2021 um 17:28:19 UTC+2:
>> On Friday, August 20, 2021 at 8:09:53 AM UTC-4,
>> WM wrote:

>>> For every [element of the set of FISONs ] a bijection
>>> with |N can be analyzed step by step
>>
>> Look! Over there! A Pink Elephant!
>> The bijection between |N and the *set of FISONs*
>> can be analyzed step by step.
>
> Only as long as the FISONs last.

The FISONs do not end.

If {0,...,k} is steppable, then {0,...,k+1} is steppable.

> Then always aleph_0 natnumbers remain not yet covered.

No natural number remains uncovered.

There is no natural number after all other natural numbers
is equivalent to
For each natural number, there is a natural number after it.

For each {0,...,k}, k+1 is not in it.

William schrieb am Samstag, 21. August 2021 um 01:26:47 UTC+2:
> On Friday, August 20, 2021 at 5:36:44 PM UTC-4, WM wrote:
> > William schrieb am Freitag, 20. August 2021 um 17:28:19 UTC+2:
> > > On Friday, August 20, 2021 at 8:09:53 AM UTC-4, WM wrote:
> > > > William schrieb am Donnerstag, 19. August 2021 um 22:48:48 UTC+2:
> > > > > On Thursday, August 19, 2021 at 2:35:52 PM UTC-4, WM wrote:
> > > > > > William schrieb am Donnerstag, 19. August 2021 um 20:18:13 UTC+2:
> > > > > > > On Thursday, August 19, 2021 at 1:53:53 PM UTC-4, WM wrote:
> > > > > > > > William schrieb am Donnerstag, 19. August 2021 um 19:24:31 UTC+2:
> > > > > > > > >
> > > > > > > you cannot use a step by step method (something you call a "well ordered way") to talk about the "completetion" of a set with no last element.
> > > > > > > > You cannot use it at all for infinite sets.
> > > > > > > Use other methods. Get results.
> > > > > > The methods you use are based upon bijections,
> > > > > Nope, Non step by step methods lead to bijections.
> > > > For every [element of the set of FISONs ] a bijection with |N can be analyzed step by step
> > >
> So what? Giving a property of every *element of set tof FISONs* tells us nothing about whether the *set of FISONs* has this property.

Therefore I talked about its elements.

> It does not. The *set of FISONs* cannot be analyzed step by step.

All FISONs can. But there is no set because it is only a potentially infinite collection

> The *set of FISONs* can be analyzed if you use the appropriate tools.

Your appropriate tools fail. Fact is that the natural number completing the indexing of all fractions of the interval (n, n+1] and the natural number completing the indexing of all fractions of the interval (n+1, n+2] cannot be distingusihed although they must be different and must be accessible by FISONs if all natnumbers have FISONs.

Regards, WM

Jim Burns schrieb am Samstag, 21. August 2021 um 15:19:25 UTC+2:
> On 8/20/2021 5:36 PM, WM wrote:

> >> The bijection between |N and the *set of FISONs*
> >> can be analyzed step by step.
> >
> > Only as long as the FISONs last.
> The FISONs do not end.

Fact is that the natural number completing the indexing of all fractions of the interval (n, n+1] and the natural number completing the indexing of all fractions of the interval (n+1, n+2] cannot be distingusihed although they must be different and must be accessible by FISONs if all natnumbers have FISONs.
>
> If {0,...,k} is steppable, then {0,...,k+1} is steppable.
> > Then always aleph_0 natnumbers remain not yet covered.
> No natural number remains uncovered.
>
> There is no natural number after all other natural numbers
> is equivalent to
> For each natural number, there is a natural number after it.

Nevertheless this does not cover all natural numbers.
>
> For each {0,...,k}, k+1 is not in it.

Correct. Nevertheless: The natural number completing the indexing of all fractions of the interval (n, n+1] and the natural number completing the indexing of all fractions of the interval (n+1, n+2] cannot be distingusihed although they must be different and must be accessible by FISONs if all natnumbers have FISONs.

Regards, WM

On 8/21/2021 8:37 AM, WM wrote:
> Jim Burns schrieb am Samstag, 21. August 2021 um 15:19:25 UTC+2:
>> On 8/20/2021 5:36 PM, WM wrote:
>
>>>> The bijection between |N and the *set of FISONs*
>>>> can be analyzed step by step.
>>>
>>> Only as long as the FISONs last.
>> The FISONs do not end.
>
> Fact is that the natural number completing the indexing of all fractions of the interval (n, n+1] and the natural number completing the indexing of all fractions of the interval (n+1, n+2] cannot be distingusihed

Sure they can, combine both intervals into one, (n,n+2] now you can natural number completing the indexing of all fractions of the interval (n,n+2]

this fact conflicts with your conclusion, so your conclusion is wrong, i.e. your thinking is wrong.

>>
>> If {0,...,k} is steppable, then {0,...,k+1} is steppable.
>>> Then always aleph_0 natnumbers remain not yet covered.
>> No natural number remains uncovered.
>>
>> There is no natural number after all other natural numbers
>> is equivalent to
>> For each natural number, there is a natural number after it.
>
> Nevertheless this does not cover all natural numbers.

>>
>> For each {0,...,k}, k+1 is not in it.
>
> Correct. Nevertheless: The natural number completing the indexing of all fractions of the interval (n, n+1] and the natural number completing the indexing of all fractions of the interval (n+1, n+2] cannot be distingusihed

Sure they can, combine both intervals into one, (n,n+2] now you can natural number completing the indexing of all fractions of the interval (n,n+2]

this fact conflicts with your conclusion, so your conclusion is wrong, i.e. your thinking is wrong.

On Saturday, August 21, 2021 at 3:37:54 PM UTC+2, WM wrote:

> > If {0,...,k} is [a FISON], then {0,...,k+1} is [a FISON].
> >
> > Then always aleph_0 natnumbers remain not yet covered.

"always" is just mumbo-jumbo, you silly crank:

Correct is that

(1) NO FISON "covers" all natural number:

~Ek e IN: An e IN: n e {1, ..., k} .

For each and every k e IN: {1, ..., k} doesn't "cover" aleph_0 natural numbers (namely the numbers k+1, k+2, k+3, ...).

(2) On the other hand:

An e IN Ek e IN: n e {1, ..., k} .

For each and every natural number n there's a FISON such that it "covers" n.

In other words, all natural numbers are "covered" (by FISONs).

> > No natural number remains uncovered.

Exactly.

> > There is no natural number after all other natural numbers is equivalent to
> > For each natural number, there is a natural number after it.>
> >
> Nevertheless this does not cover all natural numbers.

All natural numbers are "covered" (by FISONs). See explanation above.

On Saturday, August 21, 2021 at 9:35:22 AM UTC-4, WM wrote:
> William schrieb am Samstag, 21. August 2021 um 01:26:47 UTC+2:
> > On Friday, August 20, 2021 at 5:36:44 PM UTC-4, WM wrote:
> > > William schrieb am Freitag, 20. August 2021 um 17:28:19 UTC+2:
> > > > On Friday, August 20, 2021 at 8:09:53 AM UTC-4, WM wrote:
> > > > > William schrieb am Donnerstag, 19. August 2021 um 22:48:48 UTC+2:
> > > > > > On Thursday, August 19, 2021 at 2:35:52 PM UTC-4, WM wrote:
> > > > > > > William schrieb am Donnerstag, 19. August 2021 um 20:18:13 UTC+2:
> > > > > > > > On Thursday, August 19, 2021 at 1:53:53 PM UTC-4, WM wrote:
> > > > > > > > > William schrieb am Donnerstag, 19. August 2021 um 19:24:31 UTC+2:
> > > > > > > > > >
> > > > > > > > you cannot use a step by step method (something you call a "well ordered way") to talk about the "completetion" of a set with no last element.
> > > > > > > > > You cannot use it at all for infinite sets.
> > > > > > > > Use other methods. Get results.
> > > > > > > The methods you use are based upon bijections,
> > > > > > Nope, Non step by step methods lead to bijections.
> > > > > For every [element of the set of FISONs ] a bijection with |N can be analyzed step by step
> > > >
> > So what? Giving a property of every *element of set tof FISONs* tells us nothing about whether the *set of FISONs* has this property.
> Therefore I talked about its elements.
> > It does not. The *set of FISONs* cannot be analyzed step by step.
> All FISONs can. But there is no set because it is only a potentially infinite collection

Piffle. "Potentially Infinite" is nonsense.

--
William Hughes

On Saturday, August 21, 2021 at 3:35:22 PM UTC+2, WM wrote:

> But there is no set [of all FISONs] because it is only a potentially infinite collection

Look, you psychotic crank, IN SET THEORY (due to von Neumann) there *is* a set of all FISONs, it's called IN.

And right, it's an INFINITE SET.

Again, the context of this discussion is SET THEORY (say ZFC) not your psychotic belief system.

On Saturday, August 21, 2021 at 6:12:31 PM UTC+2, William wrote:

> "Potentially Infinite" is nonsense.

At least in the context of set theory (say ZFC) - for sane persons. In the context of WM's psychotic belief system it seems to "make sense" - at least for him.

William schrieb am Samstag, 21. August 2021 um 18:12:31 UTC+2:
> On Saturday, August 21, 2021 at 9:35:22 AM UTC-4, WM wrote:

> > > It does not. The *set of FISONs* cannot be analyzed step by step.
> > All FISONs can. But there is no set because it is only a potentially infinite collection
> Piffle. "Potentially Infinite" is nonsense.

Mathematics disregards your personal preferences. If you don't like potential infinity then use the natural numbers instead. They form an actually infinite set. But most can't be distinguished:

(1) When indexing the rationals all unit intervals can be completed.
(2) Not all unit intervals can be completed by one index because an index is issued to one rational of one interval only.
(3) So there are at least two indices required to complete all intervals.
(4) You cannot find out their order.

Although you prefer not to take notice of this fact, it is nevertheless unavoidable.

Regards, WM

Greg Cunt schrieb am Samstag, 21. August 2021 um 17:57:42 UTC+2:

> > Nevertheless this does not cover all natural numbers.
> All natural numbers are "covered" (by FISONs). See explanation above.

Explanations contradicted by facts are of no value.

(1) When indexing the rationals all unit intervals can be completed.
(2) Not all unit intervals can be completed by one index because an index is issued to one rational of one interval only.
(3) So there are at least two indices required to complete all intervals.
(4) You cannot find out their order by their FISONs. If you could, you would complete aleph_0 rationals of one interval but you would need further indices. Contradiction. That is the true reason for dark numbers.

Regards, WM

On Saturday, August 21, 2021 at 8:03:36 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Samstag, 21. August 2021 um 17:57:42 UTC+2:

WM claimed falsely:

> > > Nevertheless this does not cover all natural numbers.

Fact:

> > All natural numbers are "covered" (by FISONs). See explanation [...].

> Explanations contradicted by facts are of no value.

Indeed! That's why we PROVE our claims in math (mere explanations aren't good enough):

Def.: A natural number n is /covered/ iff there is a FISON F such that n e F.

In other words, a natural number n is /covered/ iff there is a natural number k such that n e {1, ..., k}.

Theorem: All natural numbers are covered.

Proof: An e IN: n e {1, ..., n}. Hence An e IN: Ek e IN: n e {1, ..., k}. qed

On Saturday, August 21, 2021 at 8:01:46 PM UTC+2, WM wrote:
> William schrieb am Samstag, 21. August 2021 um 18:12:31 UTC+2:
> > On Saturday, August 21, 2021 at 9:35:22 AM UTC-4, WM wrote:
> >
> > But there is no set [of all FISONs] because it is only a potentially infinite collection

Look, you psychotic crank, IN SET THEORY (due to von Neumann) there *is* a set of all FISONs, it's called IN.

And right, it's an INFINITE SET.

Again, the context of this discussion is SET THEORY (say ZFC) not your psychotic belief system.

> > "Potentially Infinite" is nonsense.

At least in the context of set theory (say ZFC) - for sane persons. In the context of your psychotic belief system it seems to "make sense" - at least for you.

> Mathematics disregards [.]our personal preferences.

Right. That's why we'd like to stick to (say) ZFC instead preferring your psychotic belief system.

You know: "classical mathematics refers generally to the mainstream approach to mathematics, which is based on classical logic and ZFC set theory." (Wikipedia, classical mathematics)

> If you don't like potential infinity then use the natural numbers instead. They form an actually infinite set.

Right, at least in the context of ZF(C).

> <nonsense deleted>

On 8/21/2021 9:37 AM, WM wrote:
> Jim Burns schrieb
> am Samstag, 21. August 2021 um 15:19:25 UTC+2:
>> On 8/20/2021 5:36 PM, WM wrote:

>>>> The bijection between |N and the *set of FISONs*
>>>> can be analyzed step by step.
>>>
>>> Only as long as the FISONs last.
>>
>> The FISONs do not end.
>
> Fact is that the natural number completing the indexing of
> all fractions of the interval (n, n+1]

There is no such natural number.
It would need to be a finite index after infinitely-many
indexes. There is no such natural number.

----
We can see one more iteration of this discussion,
one more time, in this little exchange.

WM --
A thing exists. (description)

JB --
Here are the things it would be one of. (description)
It isn't one of them. (reasons)

WM --
Therefore the thing I described is dark.

JB --
It isn't one of these things. (description, reasons)

WM --
Therefore your description is wrong.

....

WM --
A different thing exists. (description)

....

----
Definitions do not work the way you think they do.
A thing does not exist merely because you have a description.

That leads to nonsense, but it isn't "set-theorist" nonsense,
it's entirely your nonsense.

----
I speculate that, somewhere back in the mists of time,
you acquired a wrong idea of how "set-theorists" justify their
claims. And you noticed that _your idea_ of their justification
did not justify these claims. (And good for you for noticing,
in this scenario.)

"Set theorists" do not define things into existence in order
to justify their claims. There are basic, simple things they may
initially claim exist. And then they use truth-preserving inferences
from those initial claims to considerably less simple further
claims. "Set theorists" _do not_ pull more things out of their
magic hat in order to justify further claims.

I don't think there is a unified position on the status of the
basic, simple things initially claimed to exist. I know of two
positions.
(i) It's something Not Of This World. (Platonism)
(ii) I don't care. (formalism)

It doesn't matter which position a "set theorist" takes.
The very strong claims of mathematics are about the existence
of a sequence of truth-preserving inferences *between*
initial claims about basic things and further sometimes-bizarre
claims about the same basic things.

> and the natural number completing the indexing of all fractions
> of the interval (n+1, n+2]

Also, there is no such natural number.

> cannot be distingusihed although they must be different and
> must be accessible by FISONs if all natnumbers have FISONs.
>
>> If {0,...,k} is steppable, then {0,...,k+1} is steppable.
>>> Then always aleph_0 natnumbers remain not yet covered.
>>
>> No natural number remains uncovered.
>>
>> There is no natural number after all other natural numbers
>> is equivalent to
>> For each natural number, there is a natural number after it.
>
> Nevertheless this does not cover all natural numbers.

The numbers you think are not covered do not exist.

>> For each {0,...,k}, k+1 is not in it.
>
> Correct.
> Nevertheless: The natural number completing the indexing of
> all fractions of the interval (n, n+1] and the natural number
> completing the indexing of all fractions of the interval
> (n+1, n+2] cannot be distingusihed although they must be
> different and must be accessible by FISONs if all natnumbers
> have FISONs.

We make initial claims about a natural number, about a rational
number. That's what they are. If we change those claims, saying
that all natural numbers have FISONs -- but, no, they don't,
then we are no longer talking about what we were talking about.

It's an awesome display of incoherence. It is equivalent to my
claiming everything in {8,9,10} is composite, and you answering
and, therefore, 11 is in {8,9,10}.

On Saturday, August 21, 2021 at 2:01:46 PM UTC-4, WM wrote:
> William schrieb am Samstag, 21. August 2021 um 18:12:31 UTC+2:
> > On Saturday, August 21, 2021 at 9:35:22 AM UTC-4, WM wrote:
>
> > > > It does not. The *set of FISONs* cannot be analyzed step by step.
> > > All FISONs can. But there is no set because it is only a potentially infinite collection
> > Piffle. "Potentially Infinite" is nonsense.
> Mathematics disregards your personal preferences. If you don't like potential infinity

The reason I don't like "potential infinity" is that it is nonsense. It does not exist outside of Wolkenmuekenheim.
There is a set of FISONs, F. It is not something that changes. If x is a FISON then x is an element of F.

--
Wiliam Hughes

Crank Wolfgang Mueckenheim, aka WM wrote:
> Greg Cunt schrieb am Samstag, 21. August 2021 um 17:57:42 UTC+2:
>
>>> Nevertheless this does not cover all natural numbers.
>> All natural numbers are "covered" (by FISONs). See explanation above.
>
> Explanations contradicted by facts are of no value.
>
> (1) When indexing the rationals all unit intervals can be completed.

What does that even mean?

> (2) Not all unit intervals can be completed by one index because an index is issued to one rational of one interval only.

What does that even mean?

> (3) So there are at least two indices required to complete all intervals.

What does that even mean?

> (4) You cannot find out their order by their FISONs. If you could, you would complete aleph_0 rationals of one interval but you would need further indices. Contradiction.

What does that even mean?

> That is the true reason for dark numbers.

So the other reasons you invokes were false reasons, right?

There is no doubt that this new "reason" of yours if false also. None
of your 1-4 claims above makes any sense. NONE.

On 8/21/2021 1:01 PM, WM wrote:
> William schrieb am Samstag, 21. August 2021 um 18:12:31 UTC+2:
>> On Saturday, August 21, 2021 at 9:35:22 AM UTC-4, WM wrote:
>
>>>> It does not. The *set of FISONs* cannot be analyzed step by step.
>>> All FISONs can. But there is no set because it is only a potentially infinite collection
>> Piffle. "Potentially Infinite" is nonsense.
>
> Mathematics disregards your personal preferences.

Wrong! you WM are not "mathematics", in fact *you do not know algebra nor calculus*.

<snip made up stuff>

On 8/21/2021 1:03 PM, WM wrote:
> Greg Cunt schrieb am Samstag, 21. August 2021 um 17:57:42 UTC+2:
>
>>> Nevertheless this does not cover all natural numbers.
>> All natural numbers are "covered" (by FISONs). See explanation above.
>

<snip crap>

nothing

On 8/21/2021 1:43 PM, Jim Burns wrote:
> On 8/21/2021 9:37 AM, WM wrote:
>> Jim Burns schrieb
>> am Samstag, 21. August 2021 um 15:19:25 UTC+2:
>>> On 8/20/2021 5:36 PM, WM wrote:
>
>>>>> The bijection between |N and the *set of FISONs*
>>>>> can be analyzed step by step.
>>>>
>>>> Only as long as the FISONs last.
>>>
>>> The FISONs do not end.
>>
>> Fact is that the natural number completing the indexing of
>> all fractions of the interval (n, n+1]
>
> There is no such natural number.
> It would need to be a finite index after infinitely-many
> indexes. There is no such natural number.
>
> ----
> We can see one more iteration of this discussion,
> one more time, in this little exchange.
>
> WM --
> A thing exists. (description)
>
> JB --
> Here are the things it would be one of. (description)
> It isn't one of them. (reasons)
>
> WM --
> Therefore the thing I described is dark.
>
> JB --
> It isn't one of these things. (description, reasons)
>
> WM --
> Therefore your description is wrong.
>
> ...
>
> WM --
> A different thing exists. (description)
>
> ...
>
> ----
> Definitions do not work the way you think they do.
> A thing does not exist merely because you have a description.
>
> That leads to nonsense, but it isn't "set-theorist" nonsense,
> it's entirely your nonsense.
>
> ----
> I speculate that, somewhere back in the mists of time,
> you acquired a wrong idea of how "set-theorists" justify their
> claims. And you noticed that _your idea_ of their justification
> did not justify these claims. (And good for you for noticing,
> in this scenario.)
>
> "Set theorists" do not define things into existence in order
> to justify their claims. There are basic, simple things they may
> initially claim exist. And then they use truth-preserving inferences
> from those initial claims to considerably less simple further
> claims. "Set theorists" _do not_ pull more things out of their
> magic hat in order to justify further claims.
>
> I don't think there is a unified position on the status of the
> basic, simple things initially claimed to exist. I know of two
> positions.
> (i) It's something Not Of This World. (Platonism)
> (ii) I don't care. (formalism)
>
> It doesn't matter which position a "set theorist" takes.
> The very strong claims of mathematics are about the existence
> of a sequence of truth-preserving inferences *between*
> initial claims about basic things and further sometimes-bizarre
> claims about the same basic things.
>
>> and the natural number completing the indexing of all fractions
>> of the interval (n+1, n+2]
>
> Also, there is no such natural number.
>
>> cannot be distingusihed although they must be different and
>> must be accessible by FISONs if all natnumbers have FISONs.
>>
>>> If {0,...,k} is steppable, then {0,...,k+1} is steppable.
>>>> Then always aleph_0 natnumbers remain not yet covered.
>>>
>>> No natural number remains uncovered.
>>>
>>> There is no natural number after all other natural numbers
>>> is equivalent to
>>> For each natural number, there is a natural number after it.
>>
>> Nevertheless this does not cover all natural numbers.
>
> The numbers you think are not covered do not exist.
>
>>> For each {0,...,k}, k+1 is not in it.
>>
>> Correct.
>> Nevertheless: The natural number completing the indexing of
>> all fractions of the interval (n, n+1] and the natural number
>> completing the indexing of all fractions of the interval
>> (n+1, n+2] cannot be distingusihed although they must be
>> different and must be accessible by FISONs if all natnumbers
>> have FISONs.
>
> We make initial claims about a natural number, about a rational
> number. That's what they are. If we change those claims, saying
> that all natural numbers have FISONs -- but, no, they don't,
> then we are no longer talking about what we were talking about.
>
> It's an awesome display of incoherence. It is equivalent to my
> claiming everything in {8,9,10} is composite, and you answering
> and, therefore, 11 is in {8,9,10}.
>

New ones !!

Incoherent Ants
Incoherent Composited Ants
Illogical and Brain Dead Ants

from the Ant list 6.0

536 Incapable Step Taking Ants
537 Incompetent Irrational Complement Cluster Ants
538 Incompletness theorem by completed Ants
539 Incomprehensible Ant
540 Inconsistant Ants....
541 incorrigible asshole Ants

William schrieb am Samstag, 21. August 2021 um 20:59:41 UTC+2:
> On Saturday, August 21, 2021 at 2:01:46 PM UTC-4, WM wrote:
> > William schrieb am Samstag, 21. August 2021 um 18:12:31 UTC+2:
> > > On Saturday, August 21, 2021 at 9:35:22 AM UTC-4, WM wrote:
> >
> > > > > It does not. The *set of FISONs* cannot be analyzed step by step.
> > > > All FISONs can. But there is no set because it is only a potentially infinite collection
> > > Piffle. "Potentially Infinite" is nonsense.
> > Mathematics disregards your personal preferences. If you don't like potential infinity
> The reason I don't like "potential infinity" is that it is nonsense. It does not exist outside of Wolkenmuekenheim.

There you are very wrong.
"A potential infinity is a quantity which is finite but indefinitely large. For instance, when we enumerate the natural numbers as 0, 1, 2, ..., n, n+1, ..., the enumeration is finite at any point in time, but it grows indefinitely and without bound. [...] An actual infinity is a completed infinite totality. Examples: , , C[0, 1], L2[0, 1], etc. Other examples: gods, devils, etc." [S.G. Simpson: "Potential versus actual infinity: Insights from reverse mathematics" (2015)]

An interval of natural numbers without any prime number is called a prime gap. The sequence of prime gaps assumes arbitrarily large intervals but it cannot become actually infinite. None of the numbers n! + 2, n! + 3, n! + 4, ..., n! + n can be prime because n! = 123... n contains 2, 3, ..., n as factors already. Therefore the set of gaps has no upper bound. It is potentially infinite. It is not actually infinite however, because there does not exist a gap with no closing prime number because there is no last prime number.

Finally, the most familiar example is this: The (magnitudes of) natural numbers are potentially infinite because, although there is no upper bound, there is no infinite (magnitude of a) natural number.

> There is a set of FISONs, F. It is not something that changes. If x is a FISON then x is an element of F.

Why do you not understand facts?

(1) When indexing the rationals all unit intervals can be completed.
(2) Not all unit intervals can be completed by one index because an index is issued to one rational of one interval only.
(3) So there are at least two indices required to complete all intervals.
(4) You cannot find out their order.

Although you prefer not to take notice of this fact, it is nevertheless unavoidable.

Regards, WM

Jim Burns schrieb am Samstag, 21. August 2021 um 20:43:52 UTC+2:
> On 8/21/2021 9:37 AM, WM wrote:

> > Fact is that the natural number completing the indexing of
> > all fractions of the interval (n, n+1]
> There is no such natural number.

Then the completing is accomplished by belief only?

> It would need to be a finite index after infinitely-many
> indexes. There is no such natural number.

Completion is claimed as basis of set theory. How is it achieved?

Regards, WM

On Saturday, August 21, 2021 at 5:28:38 PM UTC-4, WM wrote:
> William schrieb am Samstag, 21. August 2021 um 20:59:41 UTC+2:
> > On Saturday, August 21, 2021 at 2:01:46 PM UTC-4, WM wrote:
> > > William schrieb am Samstag, 21. August 2021 um 18:12:31 UTC+2:
> > > > On Saturday, August 21, 2021 at 9:35:22 AM UTC-4, WM wrote:
> > >
> > > > > > It does not. The *set of FISONs* cannot be analyzed step by step.
> > > > > All FISONs can. But there is no set because it is only a potentially infinite collection
> > > > Piffle. "Potentially Infinite" is nonsense.
> > > Mathematics disregards your personal preferences. If you don't like potential infinity
> > The reason I don't like "potential infinity" is that it is nonsense. It does not exist outside of Wolkenmuekenheim.
> There you are very wrong.
> "A potential infinity is a quantity which is finite but indefinitely large. For instance, when we enumerate the natural numbers as 0, 1, 2, ..., n, n+1, ..., the enumeration is finite at any point in time, but it grows indefinitely and without bound. [...] An actual infinity is a completed infinite totality.
>

The use of the word "completed" gives this away. It is nonsense. It changes nothing.

--
William Hughes

Python schrieb am Samstag, 21. August 2021 um 21:34:11 UTC+2:
> WM wrote:

> > (1) When indexing the rationals all unit intervals can be completed.
> What does that even mean?

That can be answered best by the master himself (the German terminus is vollständig):

"Zwei wohlgeordnete Mengen M und N heissen von gleichem Typus oder auch von gleicher Anzahl, wenn sie sich gegenseitig eindeutig und vollständig unter beidseitiger Wahrung der Rangfolge ihrer Elemente auf einander beziehen, abbilden lassen;" [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]

"Zwei bestimmte Mengen M und M1 nennen wir äquivalent (in Zeichen: M ~ M1), wenn es möglich ist, dieselben gesetzmäßig, gegenseitig eindeutig und vollständig, Element für Element, einander zuzuordnen." [Cantor, p. 412]

"doch gibt es immer viele, im allgemeinen sogar unzählig viele Zuordnungsgesetze, durch welche zwei äquivalente Mengen in gegenseitig eindeutige und vollständige Beziehung zueinander gebracht werden können.." [Cantor, p. 413]

"eine solche gegenseitig eindeutige und vollständige Korrespondenz hergestellt [...] irgendeine gegenseitig eindeutige und vollständige Zuordnung der beiden Mengen [...] auch eine gegenseitig eindeutige und vollständige Korrespondenz" [Cantor, p. 415]

"Zwei n-fach geordnete Mengen M und N werden 'ähnlich' genannt, wenn es möglich ist, sie gegenseitig eindeutig und vollständig, Element für Element, einander so zuzuordnen," [Cantor, p. 424]

> > (2) Not all unit intervals can be completed by one index because an index is issued to one rational of one interval only.
> What does that even mean?

A rational number belongs to one and only one unit interval.

Regards, WM