zelos...@gmail.com schrieb am Montag, 29. November 2021 um 12:51:53 UTC+1:
> måndag 29 november 2021 kl. 11:28:00 UTC+1 skrev WM:
> > zelos...@gmail.com schrieb am Montag, 29. November 2021 um 06:51:36 UTC+1:
> > > fredag 26 november 2021 kl. 17:24:10 UTC+1 skrev WM:
> > > > William schrieb am Freitag, 26. November 2021 um 17:06:56 UTC+1:
> > > > It cannot have same numbers of elements as {0} U ℕ, let alone be in bijection, i.e., in 1-1-correspondence with |Q.
> > > >
> > > who says the size of the set changes? it doesn't!
> > >
> > > But it can sitll be in bijection with those sets and all be of the same size!
> > >
> > > It is a counterintuitive result but very valid!
> > It is a clear contradiction because bijections cause equal *number*. ℕ and {0} U ℕ differ by one number.
> >
> Nope, you think that because it is how it is for FINITE sets but that is NOT the case for INFINITE sets.

Bijection means bijection and has nowhere another meaning.
>
> Infinite sets CAN be set into a bijection with a PROPER subset, DEFINITIONALLY.

That is true for potentially infinite sets, made by Dedekind.
>
> That is what defined them and is NOT a contradiction!

That has been confused by some fools.

Regards, WM

FromTheRafters schrieb am Montag, 29. November 2021 um 12:53:58 UTC+1:
> WM expressed precisely :
> > zelos...@gmail.com schrieb am Montag, 29. November 2021 um 06:51:36 UTC+1:
> >> fredag 26 november 2021 kl. 17:24:10 UTC+1 skrev WM:
> >>> William schrieb am Freitag, 26. November 2021 um 17:06:56 UTC+1:
> >>> It cannot have same numbers of elements as {0} U ℕ, let alone be in
> >>> bijection, i.e., in 1-1-correspondence with |Q.
> >>>
> >> who says the size of the set changes? it doesn't!
> >>
> >> But it can sitll be in bijection with those sets and all be of the same
> >> size!
> >>
> >> It is a counterintuitive result but very valid!
> >
> > It is a clear contradiction because bijections cause equal *number*.
> With finite sets, yes, more like equal *size* when dealing with
> infinite sets.

No, bijection means bijection in every case. Only potential infinity can show Dedekind-infiniteness.

Regards, WM

On 11/29/2021 4:30 AM, WM wrote:
> William schrieb am Sonntag, 28. November 2021 um 17:31:25 UTC+1:
>> On Sunday, November 28, 2021 at 6:18:12 AM UTC-5, WM wrote:
>>> . ..for all Peano elements we have
>>> ∀n ∈ ℕ_P: |ℕ \ {1, 2, 3, ..., n}| = ℵ₀.
>>> There is no way to reduce the remaining set of ℵ₀ to less than ℵ₀ by removing individual natnumbers.
>> Since by "removing individual natnumbers" you mean removing a finite set, this is correct.
>
> It suffices to show that the definable numbers are less than ℵ₀.

prove it. you are just making stuff up.

>
>> You need to remove a Peano set which has cardinality aleph_0.
>
> The definable numbers are less.

prove it. your instantated, defined number daffynition is silly, not math.

>
>> However, when you remove the Peano set you do not remove any "dark" elements.
>>
> Individually many can be removed but most (ℵ₀) cannot be removed. That is what I call dark.

dark only to you.

>
> Regards, WM
>

On Monday, November 29, 2021 at 5:30:59 AM UTC-5, WM wrote:
>definable numbers
are an irrelevant entity. Of interest is the Peano set of natural numbers.

--William Hughes

WM brought next idea :
> zelos...@gmail.com schrieb am Montag, 29. November 2021 um 12:51:53 UTC+1:
>> måndag 29 november 2021 kl. 11:28:00 UTC+1 skrev WM:
>>> zelos...@gmail.com schrieb am Montag, 29. November 2021 um 06:51:36 UTC+1:
>>>> fredag 26 november 2021 kl. 17:24:10 UTC+1 skrev WM:
>>>>> William schrieb am Freitag, 26. November 2021 um 17:06:56 UTC+1:
>>>>> It cannot have same numbers of elements as {0} U ℕ, let alone be in
>>>>> bijection, i.e., in 1-1-correspondence with |Q.
>>>>>
>>>> who says the size of the set changes? it doesn't!
>>>>
>>>> But it can sitll be in bijection with those sets and all be of the same
>>>> size!
>>>>
>>>> It is a counterintuitive result but very valid!
>>> It is a clear contradiction because bijections cause equal *number*. ℕ and
>>> {0} U ℕ differ by one number.
>>>
>> Nope, you think that because it is how it is for FINITE sets but that is NOT
>> the case for INFINITE sets.
>
> Bijection means bijection and has nowhere another meaning.

In this context it is a comparison of a notion of size, not of number.
That there is a bijection only implies that they are comparable in
size. This is borne out in the finite case by there being the same
number of elements.

>> Infinite sets CAN be set into a bijection with a PROPER subset,
>> DEFINITIONALLY.
>
> That is true for potentially infinite sets, made by Dedekind.
>>
>> That is what defined them and is NOT a contradiction!
>
> That has been confused by some fools.

You are confused, so you are insulting yourself.

WM presented the following explanation :
> FromTheRafters schrieb am Montag, 29. November 2021 um 12:53:58 UTC+1:
>> WM expressed precisely :
>>> zelos...@gmail.com schrieb am Montag, 29. November 2021 um 06:51:36 UTC+1:
>>>> fredag 26 november 2021 kl. 17:24:10 UTC+1 skrev WM:
>>>>> William schrieb am Freitag, 26. November 2021 um 17:06:56 UTC+1:
>>>>> It cannot have same numbers of elements as {0} U ℕ, let alone be in
>>>>> bijection, i.e., in 1-1-correspondence with |Q.
>>>>>
>>>> who says the size of the set changes? it doesn't!
>>>>
>>>> But it can sitll be in bijection with those sets and all be of the same
>>>> size!
>>>>
>>>> It is a counterintuitive result but very valid!
>>>
>>> It is a clear contradiction because bijections cause equal *number*.
>> With finite sets, yes, more like equal *size* when dealing with
>> infinite sets.
>
> No, bijection means bijection in every case.

Duh!

William schrieb am Montag, 29. November 2021 um 15:04:29 UTC+1:
> On Monday, November 29, 2021 at 5:30:59 AM UTC-5, WM wrote:
> >definable numbers
> are an irrelevant entity. Of interest is the Peano set of natural numbers..

Numbers that cannot be applied individually are of no interest whatsoever.
Most natural numbers cannot be applied as individuals since for all such individually definable numbers
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo ,
hence the set of undefinable numbers has ℵo elements
and two consecutive ℵo-sets in the natural order of are ℕ are impossible.

Regards, WM

FromTheRafters schrieb am Montag, 29. November 2021 um 15:15:29 UTC+1:
> WM presented the following explanation :
> > FromTheRafters schrieb am Montag, 29. November 2021 um 12:53:58 UTC+1:

> >>>> who says the size of the set changes? it doesn't!
> >>>>
> >>>> But it can sitll be in bijection with those sets and all be of the same
> >>>> size!
> >>>>
> >>>> It is a counterintuitive result but very valid!
> >>>
> >>> It is a clear contradiction because bijections cause equal *number*.
> >> With finite sets, yes, more like equal *size* when dealing with
> >> infinite sets.
> >
> > No, bijection means bijection in every case.
> Duh!

One-to-one! Unfortunately set theory has to pervert mathematics and the thinking of mathematicians in order to survive. Surprising is only its splendid success.

Regards, WM

On Monday, November 29, 2021 at 4:11:52 PM UTC+1, WM wrote:
> FromTheRafters schrieb am Montag, 29. November 2021 um 15:15:29 UTC+1:
> > WM presented the following explanation :
> > > FromTheRafters schrieb am Montag, 29. November 2021 um 12:53:58 UTC+1:
>
> > >>>> who says the size of the set changes? it doesn't!
> > >>>>
> > >>>> But it can sitll be in bijection with those sets and all be of the same
> > >>>> size!
> > >>>>
> > >>>> It is a counterintuitive result but very valid!
> > >>>
> > >>> It is a clear contradiction because bijections cause equal *number*..
> > >> With finite sets, yes, more like equal *size* when dealing with
> > >> infinite sets.
> > >
> > > No, bijection means bijection in every case.
> > Duh!
> One-to-one! Unfortunately set theory has to pervert mathematics and the thinking of mathematicians in order to survive. Surprising is only its splendid success.
>
> Regards, WM
Evidently Mückenheim's head is in permanent bijection with his arse.
That's where he discovered the dark proctological numbers.

On 11/29/2021 9:10 AM, WM wrote:
> William schrieb am Montag, 29. November 2021 um 15:04:29 UTC+1:
>> On Monday, November 29, 2021 at 5:30:59 AM UTC-5, WM wrote:
>>> definable numbers
>> are an irrelevant entity. Of interest is the Peano set of natural numbers.
>
> Numbers that cannot be applied individually are of no interest whatsoever.
> Most natural numbers cannot be applied as individuals since for all such individually definable numbers
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo ,
> hence the set of undefinable numbers has ℵo elements
> and two consecutive ℵo-sets in the natural order of are ℕ are impossible.
>
> Regards, WM
>

Ants that cannot be applied individually are of no interest whatsoever.

the set of undefinable Ants has ℵo Ants

two consecutive Ants in line are impossible.

On 11/29/2021 9:11 AM, WM wrote:
> FromTheRafters schrieb am Montag, 29. November 2021 um 15:15:29 UTC+1:
>> WM presented the following explanation :
>>> FromTheRafters schrieb am Montag, 29. November 2021 um 12:53:58 UTC+1:
>
>>>>>> who says the size of the set changes? it doesn't!
>>>>>>
>>>>>> But it can sitll be in bijection with those sets and all be of the same
>>>>>> size!
>>>>>>
>>>>>> It is a counterintuitive result but very valid!
>>>>>
>>>>> It is a clear contradiction because bijections cause equal *number*.
>>>> With finite sets, yes, more like equal *size* when dealing with
>>>> infinite sets.
>>>
>>> No, bijection means bijection in every case.
>> Duh!
>
> One-to-one! Unfortunately set theory has to pervert mathematics and the thinking of mathematicians in order to survive. Surprising is only its splendid success.
>
> Regards, WM
>

with such a view of Math, no wonder you try to smear it with your falsehoods in every single post.
you hate math!

On Monday, November 29, 2021 at 10:10:58 AM UTC-5, WM wrote:

> Most natural numbers cannot be applied as individuals

"applied as individuals" is another way you have of saying "cannot be written down". Numbers that cannot be written down still have other properties and are of interest. The fact that most elements of a Peano set (like the natural numbers) cannot be written down is a triviallity.

-- William Hughes

On 11/29/2021 11:03 AM, William wrote:
> On Monday, November 29, 2021 at 10:10:58 AM UTC-5, WM wrote:
>
>
>
>> Most natural numbers cannot be applied as individuals
>
> "applied as individuals" is another way you have of saying "cannot be written down". Numbers that cannot be written down still have other properties and are of interest. The fact that most elements of a Peano set (like the natural numbers) cannot be written down is a triviallity.
>
> -- William Hughes
>
>

this is obvious;

Individual Numbers must apply to be a member of the set of Natural Numbers.

wonder what they put down as their name on the application form ?

It happens that WM formulated :
> FromTheRafters schrieb am Montag, 29. November 2021 um 15:15:29 UTC+1:
>> WM presented the following explanation :
>>> FromTheRafters schrieb am Montag, 29. November 2021 um 12:53:58 UTC+1:
>
>>>>>> who says the size of the set changes? it doesn't!
>>>>>>
>>>>>> But it can sitll be in bijection with those sets and all be of the same
>>>>>> size!
>>>>>>
>>>>>> It is a counterintuitive result but very valid!
>>>>>
>>>>> It is a clear contradiction because bijections cause equal *number*.
>>>> With finite sets, yes, more like equal *size* when dealing with
>>>> infinite sets.
>>>
>>> No, bijection means bijection in every case.
>> Duh!
>
> One-to-one!

The first one of this set gets paired with the first one of the other
set, then the next and the next and the next etcetera.

måndag 29 november 2021 kl. 14:36:50 UTC+1 skrev WM:
> zelos...@gmail.com schrieb am Montag, 29. November 2021 um 12:51:53 UTC+1:
> > måndag 29 november 2021 kl. 11:28:00 UTC+1 skrev WM:
> > > zelos...@gmail.com schrieb am Montag, 29. November 2021 um 06:51:36 UTC+1:
> > > > fredag 26 november 2021 kl. 17:24:10 UTC+1 skrev WM:
> > > > > William schrieb am Freitag, 26. November 2021 um 17:06:56 UTC+1:
> > > > > It cannot have same numbers of elements as {0} U ℕ, let alone be in bijection, i.e., in 1-1-correspondence with |Q.
> > > > >
> > > > who says the size of the set changes? it doesn't!
> > > >
> > > > But it can sitll be in bijection with those sets and all be of the same size!
> > > >
> > > > It is a counterintuitive result but very valid!
> > > It is a clear contradiction because bijections cause equal *number*. ℕ and {0} U ℕ differ by one number.
> > >
> > Nope, you think that because it is how it is for FINITE sets but that is NOT the case for INFINITE sets.
> Bijection means bijection and has nowhere another meaning.
> >
> > Infinite sets CAN be set into a bijection with a PROPER subset, DEFINITIONALLY.
> That is true for potentially infinite sets, made by Dedekind.
> >
> > That is what defined them and is NOT a contradiction!
> That has been confused by some fools.
>
> Regards, WM

> Bijection means bijection and has nowhere another meaning.

Correct, and I can put N and Z into bijection with each other :) So same cardinality!

> That is true for potentially infinite sets, made by Dedekind.

It is true for infinite sets. Your "potential"/"actual" garbage has no place in mathematics.

> That has been confused by some fools.

Mostly you, infinite sets is confusing to you

zelos...@gmail.com schrieb am Dienstag, 30. November 2021 um 06:45:19 UTC+1:
> måndag 29 november 2021 kl. 14:36:50 UTC+1 skrev WM:

> > Bijection means bijection and has nowhere another meaning.
> Correct, and I can put N and Z into bijection with each other :) So same cardinality!

You can't. All you can belongs to ℕ_def but
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
You cannot remove the huge set of ℵo remainings.

> > That is true for potentially infinite sets, made by Dedekind.
> It is true for infinite sets. Your "potential"/"actual" garbage has no place in mathematics.

say the liars and the stupids. Are you proud to belong to the gang of counter-mathematicians?

Cantor emphasized plainly and constantly that all transfinite objects of his set theory are based on the actual infinity. Modern AST-people try to persuade us to believe that the AST does not use actual infinity. It is an intentional and blatant lie [A. Zenkin, letter to D. Zeilberger (20 Dec 2005)]

Regards, WM

tisdag 30 november 2021 kl. 13:33:12 UTC+1 skrev WM:
> zelos...@gmail.com schrieb am Dienstag, 30. November 2021 um 06:45:19 UTC+1:
> > måndag 29 november 2021 kl. 14:36:50 UTC+1 skrev WM:
>
> > > Bijection means bijection and has nowhere another meaning.
> > Correct, and I can put N and Z into bijection with each other :) So same cardinality!
> You can't. All you can belongs to ℕ_def but
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> You cannot remove the huge set of ℵo remainings.
> > > That is true for potentially infinite sets, made by Dedekind.
> > It is true for infinite sets. Your "potential"/"actual" garbage has no place in mathematics.
> say the liars and the stupids. Are you proud to belong to the gang of counter-mathematicians?
>
> Cantor emphasized plainly and constantly that all transfinite objects of his set theory are based on the actual infinity. Modern AST-people try to persuade us to believe that the AST does not use actual infinity. It is an intentional and blatant lie [A. Zenkin, letter to D. Zeilberger (20 Dec 2005)]
>
> Regards, WM

>You can't. All you can belongs to ℕ_def but

False, I can with N

f:N->Z
if x is even, f(x)=x/2
if x is odd, f(x)=-(x+1)/2

>∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

making N_def=N

>You cannot remove the huge set of ℵo remainings.

all elements are in N, none remaining :)

>say the liars and the stupids.

nope, says mathematicians. You are not a mathematician

>Are you proud to belong to the gang of counter-mathematicians?

I belong to no such. I deal with actual mathematics, real mathematics. Unlike you. You are a crank that do not understand mathematics.

>Cantor emphasized plainly and constantly that all transfinite objects of his set theory are based on the actual infinity. Modern AST-people try to persuade us to believe that the AST does not use actual infinity. It is an intentional and blatant lie [A. Zenkin, letter to D. Zeilberger (20 Dec 2005)]

Zeilberger is an ultrafinitst crank, so who cares?

William schrieb am Montag, 29. November 2021 um 18:03:21 UTC+1:
> On Monday, November 29, 2021 at 10:10:58 AM UTC-5, WM wrote:
>
>
>
> > Most natural numbers cannot be applied as individuals
> "applied as individuals" is another way you have of saying "cannot be written down".

No "applied as individuals" means "can be written down".

> Numbers that cannot be written down still have other properties and are of interest.

That may be but they cannot be applied as individuals.

> The fact that most elements of a Peano set (like the natural numbers) cannot be written down is a triviallity.

Yes it is trivial, but not yet understood by most who claim that all natural numbers can be applied as individuals or written down.

Regards, WM

zelos...@gmail.com schrieb am Dienstag, 30. November 2021 um 13:48:23 UTC+1:
> tisdag 30 november 2021 kl. 13:33:12 UTC+1 skrev WM:

> >You can't. All you can belongs to ℕ_def but
> False, I can with N
>
> f:N->Z
> if x is even, f(x)=x/2
> if x is odd, f(x)=-(x+1)/2
> >∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> making N_def=N

That is no individual definition. Note decimal representation!

> >You cannot remove the huge set of ℵo remainings.
> all elements are in N, none remaining :)

Note decimal representation!

Regards, WM

On 11/30/2021 6:33 AM, WM wrote:
> zelos...@gmail.com schrieb am Dienstag, 30. November 2021 um 06:45:19 UTC+1:
>> måndag 29 november 2021 kl. 14:36:50 UTC+1 skrev WM:
>
>>> Bijection means bijection and has nowhere another meaning.
>> Correct, and I can put N and Z into bijection with each other :) So same cardinality!
>
> You can't. All you can belongs to ℕ_def but
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> You cannot remove the huge set of ℵo remainings.
>
>>> That is true for potentially infinite sets, made by Dedekind.
>> It is true for infinite sets. Your "potential"/"actual" garbage has no place in mathematics.
>
> say the liars and the stupids. Are you proud to belong to the gang of counter-mathematicians?
>
> Cantor emphasized plainly and constantly that all transfinite objects of his set theory are based on the actual infinity. Modern AST-people try to persuade us to believe that the AST does not use actual infinity. It is an intentional and blatant lie [A. Zenkin, letter to D. Zeilberger (20 Dec 2005)]
>
> Regards, WM
>

you are the one out of mathematics, instead of standard math books you use some off the wall dudes letter from 16 years ago, silly.

On Tuesday, November 30, 2021 at 7:51:37 AM UTC-5, WM wrote:
> William schrieb am Montag, 29. November 2021 um 18:03:21 UTC+1:

> > Numbers that cannot be written down still have other properties
> That may be

So elements of the Peano set of natural numbers that cannot be written down still have other properties(e.g., an element of the set of natural numbers is the largest element of a FISON)

--
William Hughes

William schrieb am Dienstag, 30. November 2021 um 16:55:27 UTC+1:
> On Tuesday, November 30, 2021 at 7:51:37 AM UTC-5, WM wrote:
> > William schrieb am Montag, 29. November 2021 um 18:03:21 UTC+1:
>
> > > Numbers that cannot be written down still have other properties
> > That may be
>
> So elements of the Peano set of natural numbers that cannot be written down still have other properties(e.g., an element of the set of natural numbers is the largest element of a FISON)

But all Peano numbers or largest elements of FISONs are followed by ℵ₀ dark elements. So the set of numbers followed by ℵ₀ numbers cannot have cardinality ℵ₀.

Regards, WM

On Tuesday, November 30, 2021 at 4:22:15 PM UTC-5, WM wrote:
> William schrieb am Dienstag, 30. November 2021 um 16:55:27 UTC+1:
> > On Tuesday, November 30, 2021 at 7:51:37 AM UTC-5, WM wrote:
> > > William schrieb am Montag, 29. November 2021 um 18:03:21 UTC+1:
> >
> > > > Numbers that cannot be written down still have other properties
> > > That may be
> >
> > So elements of the Peano set of natural numbers that cannot be written down still have other properties(e.g., an element of the set of natural numbers is the largest element of a FISON)
> But all Peano numbers or largest elements of FISONs are followed by ℵ₀ dark elements.
Nope, they are followed by a set of natural numbers with cardinality ℵ₀ (not the same set each time). The Peano set of natural numbers does not contain "dark" elements.

--
William Hughes

tisdag 30 november 2021 kl. 14:01:43 UTC+1 skrev WM:
> zelos...@gmail.com schrieb am Dienstag, 30. November 2021 um 13:48:23 UTC+1:
> > tisdag 30 november 2021 kl. 13:33:12 UTC+1 skrev WM:
>
> > >You can't. All you can belongs to ℕ_def but
> > False, I can with N
> >
> > f:N->Z
> > if x is even, f(x)=x/2
> > if x is odd, f(x)=-(x+1)/2
> > >∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> > making N_def=N
> That is no individual definition. Note decimal representation!
> > >You cannot remove the huge set of ℵo remainings.
> > all elements are in N, none remaining :)
> Note decimal representation!
>
> Regards, WM

What about it? It is of no importance as it is just representation.

tisdag 30 november 2021 kl. 22:22:15 UTC+1 skrev WM:
> William schrieb am Dienstag, 30. November 2021 um 16:55:27 UTC+1:
> > On Tuesday, November 30, 2021 at 7:51:37 AM UTC-5, WM wrote:
> > > William schrieb am Montag, 29. November 2021 um 18:03:21 UTC+1:
> >
> > > > Numbers that cannot be written down still have other properties
> > > That may be
> >
> > So elements of the Peano set of natural numbers that cannot be written down still have other properties(e.g., an element of the set of natural numbers is the largest element of a FISON)
> But all Peano numbers or largest elements of FISONs are followed by ℵ₀ dark elements. So the set of numbers followed by ℵ₀ numbers cannot have cardinality ℵ₀.
>
> Regards, WM
you have yet to define what "dark" means that doesn't make the set of them empty.