On 4/25/24 3:51 PM, WM wrote:
> Le 24/04/2024 à 20:08, FromTheRafters a écrit :
>> WM explained :
>>> Le 24/04/2024 à 12:49, FromTheRafters a écrit :
>>>> Richard Damon wrote :
>>>
>>>>> Of course, if you logic tries to make that transfinite value finite
>>>>> because that is all it can handle, your system just blows up in
>>>>> your face.
>>>>
>>>> IMO, it might be best not to think of it as a value at all.
>>>
>>> Cantor called it a number.
>>
>> A number representing an order type.
>
> like 2 or 5.
>
> Regards, WM

No, because 2 or 5 were FINITE numbers, not transfinite numbers.

Transfinite numbers act differently if they are ordinals or cardinal
numbers, which your logic just doesn't understand.

On 4/25/2024 1:27 PM, WM wrote:
> Le 25/04/2024 à 21:47, Moebius a écrit :
>> Am 25.04.2024 um 21:40 schrieb WM:
>>> Le 24/04/2024 à 19:40, Moebius a écrit :
>>>> Am 24.04.2024 um 19:01 schrieb WM:
>>>>
>>>>> What is between ω and ω*2?
>>>>
>>>> Was wohl, Du Hirni? Wenn wir von Ordinalzahlen reden, dann gilt:
>>>>
>>>> ω < ω+1 < ω+2 < ... < ω+ω = ω*2
>>>>
>>>> Die Menge der Ordinalzahlen, die zwischen ω und ω*2 liegen, ist also
>>>>
>>>> {o e ORD : ω < o < ω*2} = {ω+1, ω+2, ω+2, ...}.
>>>>
>>>> Übrigens enthält diese Menge aleph_0 Ordinalzahlen
>>>
>>> every second of which is in [...] (0, ω)*2.
>>
>> None of which is in (0, ω)*2, since (0, ω)*2 c (0, ω)
>
> Nonsense.
> The complete set is enlarged by doubling.

Sigh... Wow.

On 4/25/2024 3:14 PM, Moebius wrote:
> Am 25.04.2024 um 21:23 schrieb Chris M. Thomasson:
>
> > what about:
> >
> > [0] = 0
> > [1] = 1/3
> > [2] = 2/3
> > [3] = 1
>
> Yes. Now look at Cantor's approach (if you don't know it already):
>
> [1] = 1/1
>
> [2] = 1/2
> [3] = 2/1
>
> [4] = 1/3
> [5] = 2/2
> [6] = 3/1
>
> [7] = 1/4
> [8] = 2/3
> [9] = 3/2
> [10]= 4/1
>
>  :
>
> This shows that the (positive) fractions are "countable". (This implies
> that the rational numbers are "countable" or [maybe a better term]
> "enumerable".)
>
> > .(3) for 1/3 is just how base 10 represents 1/3. Think about hitting
> a cycle during any long division in base 10. We can stop the process at
> any detected "cycle"... Once we hit a cycle its a rational. Fair enough?
> Or did I fuck up again?
>
> No. We can indeed prove that a real number is rational iff its decimal
> expansion contains such a cycle.

Indeed. Fwiw, here is some experimental MIDI music I made using Cantor
Pairing. It shows some of my older plots:

https://youtu.be/XkwgJt5bxKI

It was fun to create. :^)

I suppose Cantor Pairing is another one that WM does not understand. We
can map back-and-forth using the pairing functions. Nothing is lost, and
nothing is dark. WM is insane, or a _very_ strange troll, or both? ;^o

Richard Damon schrieb:
> On 4/25/24 4:15 PM, WM wrote:
>> Le 25/04/2024 à 01:13, Richard Damon a écrit :
>>> On 4/24/24 1:05 PM, WM wrote:
>>
>>>> Is there an ordinal between ℕ and ω? If not, then the next number
>>>> below ω is a natural.
>>>
>>> Nope, because "next below" is not an operation defined for ω, just
>>> like it isn't for 0.
>>
>> Defined or not, there is something on the way from 1 to ω. Only natural
>> numbers or empty steps.
>
> But there is no "step" to ω from the value before it, only the hyper
> step from the full set of finite numbers.
>
>>>
>>> The ONLY numbers below ω are the Natural Numbers, but "next" doesn't
>>> apply.
>>
>> If ω is existing, then something is next.
>
> next above, yes, ω+1
>
> next below, no, the problem is the set below it is unbounded, and thus
> doesn't have a "highest" value in it.

Better say something like "the case" but don't say "the problem", because WM
takes this as an admission that today's set theory is "unlogical" and wrong.
;)

On 4/26/24 7:14 AM, Tom Bola wrote:
> Richard Damon schrieb:
>> On 4/25/24 4:15 PM, WM wrote:
>>> Le 25/04/2024 à 01:13, Richard Damon a écrit :
>>>> On 4/24/24 1:05 PM, WM wrote:
>>>
>>>>> Is there an ordinal between ℕ and ω? If not, then the next number
>>>>> below ω is a natural.
>>>>
>>>> Nope, because "next below" is not an operation defined for ω, just
>>>> like it isn't for 0.
>>>
>>> Defined or not, there is something on the way from 1 to ω. Only natural
>>> numbers or empty steps.
>>
>> But there is no "step" to ω from the value before it, only the hyper
>> step from the full set of finite numbers.
>>
>>>>
>>>> The ONLY numbers below ω are the Natural Numbers, but "next" doesn't
>>>> apply.
>>>
>>> If ω is existing, then something is next.
>>
>> next above, yes, ω+1
>>
>> next below, no, the problem is the set below it is unbounded, and thus
>> doesn't have a "highest" value in it.
>
> Better say something like "the case" but don't say "the problem", because WM
> takes this as an admission that today's set theory is "unlogical" and wrong.
> ;)

But the "Problem" is just in his logic. HE is the one that want there to
be a predecessor to ω, today's set theory doesn't make that claim. There
is ORDER between the parts of the set, but not a "sequence"

There *IS* an "unlogic" in the statement, but only because HE tries to
make something that isn't.

Le 26/04/2024 à 01:03, FromTheRafters a écrit :
> After serious thinking WM wrote :
>> Le 24/04/2024 à 20:08, FromTheRafters a écrit :
>>> WM explained :
>>>> Le 24/04/2024 à 12:49, FromTheRafters a écrit :
>>>>> Richard Damon wrote :
>>>>
>>>>>> Of course, if you logic tries to make that transfinite value finite
>>>>>> because that is all it can handle, your system just blows up in your
>>>>>> face.
>>>>>
>>>>> IMO, it might be best not to think of it as a value at all.
>>>>
>>>> Cantor called it a number.
>>>
>>> A number representing an order type.
>>
>> like 2 or 5.
>
> No, like:

Read Cantor. It is not difficult. Even you can understand it.
Jeder geordneten Menge M kommt ein bestimmter "Ordnungstypus" oder kürzer
ein bestimmter "Typus" zu.

Regards, WM

Le 26/04/2024 à 01:11, Jim Burns a écrit :
> On 4/25/2024 4:03 PM, WM wrote:

>> If all smaller numbers are doubled,
>> then there is no place for
>> the doubled numbers below ω.
>
> If n is below ω
> then n can be counted to from 0
> then n⋅2 can be counted to from n

That is true for definable numbers but not for the last numbers before ω.

>>> I defined a map for which ω+3 ⟼ 2⋅3

To my knowledge you did not. And if so, why did you?

Regards, WM

Le 26/04/2024 à 04:03, Richard Damon a écrit :

>> If ω is existing, then something is next.
>
> next above, yes, ω+1
>
> next below, no,

I disagree.

> the problem is the set below it is unbounded,

The set is bounded by ω.

> and thus
> doesn't have a "highest" value in it.

The value is there but invisible.

Regards, WM

On 4/26/24 10:41 AM, WM wrote:
> Le 26/04/2024 à 04:03, Richard Damon a écrit :
>
>>> If ω is existing, then something is next.
>>
>> next above, yes, ω+1
>>
>> next below, no,
>
> I disagree.

Which makes you wrong.

>
>> the problem is the set below it is unbounded,
>
> The set is bounded by ω.

But ω isn't in the set, so it can't be the upper bound of the set that
is below ω

Your logic just said that ω is the number before ω, so got stuck in a
infinite loop.

After all, if the number below ω is the upper bound of the set of
Natural Numbers, and that is itself ω, then the number below ω must be ω
by your logic.

BOOM.

>
>> and thus doesn't have a "highest" value in it.
>
> The value is there but invisible.
>

Nope. No such thing.

> Regards, WM
>
>

Richard Damon schrieb:
> On 4/26/24 7:14 AM, Tom Bola wrote:
>> Richard Damon schrieb:
>>> On 4/25/24 4:15 PM, WM wrote:
>>>> Le 25/04/2024 à 01:13, Richard Damon a écrit :
>>>>> On 4/24/24 1:05 PM, WM wrote:
>>>>
>>>>>> Is there an ordinal between ℕ and ω? If not, then the next number
>>>>>> below ω is a natural.
>>>>>
>>>>> Nope, because "next below" is not an operation defined for ω, just
>>>>> like it isn't for 0.
>>>>
>>>> Defined or not, there is something on the way from 1 to ω. Only natural
>>>> numbers or empty steps.
>>>
>>> But there is no "step" to ω from the value before it, only the hyper
>>> step from the full set of finite numbers.
>>>
>>>>>
>>>>> The ONLY numbers below ω are the Natural Numbers, but "next" doesn't
>>>>> apply.
>>>>
>>>> If ω is existing, then something is next.
>>>
>>> next above, yes, ω+1
>>>
>>> next below, no, the problem is the set below it is unbounded, and thus
>>> doesn't have a "highest" value in it.
>>
>> Better say something like "the case" but don't say "the problem", because WM
>> takes this as an admission that today's set theory is "unlogical" and wrong.
>> ;)
>
> But the "Problem" is just in his logic. HE is the one that want there to
> be a predecessor to ω, today's set theory doesn't make that claim. There
> is ORDER between the parts of the set, but not a "sequence"
>
> There *IS* an "unlogic" in the statement, but only because HE tries to
> make something that isn't.

Right, "the problem" is not our definitions of the set theory but WMs
belief that "the mathematics" is "realized" within "Nature" in the way
normal people can "see" - and any other "opinion" is "unlogical" and it
is even crime to teach the wrong "theory" of the "Matheologicans" (which
are all "believers" of todays math) to children within any educational
system.

Le 26/04/2024 à 16:49, Richard Damon a écrit :
> On 4/26/24 10:41 AM, WM wrote:

>>
>> The set is bounded by ω.
>
> But ω isn't in the set, so it can't be the upper bound of the set that
> is below ω

Learn the meaning of upper bound. ω*2 is an upper bound too.

Fact is that below ω there are natnumbers and above there are none. Hence
they cease below the bound ω.

Regards, WM

Le 26/04/2024 à 04:08, Richard Damon a écrit :
> On 4/25/24 3:51 PM, WM wrote:
>> Le 24/04/2024 à 20:08, FromTheRafters a écrit :
>>> WM explained :
>>>> Le 24/04/2024 à 12:49, FromTheRafters a écrit :
>>>>> Richard Damon wrote :
>>>>
>>>>>> Of course, if you logic tries to make that transfinite value finite
>>>>>> because that is all it can handle, your system just blows up in
>>>>>> your face.
>>>>>
>>>>> IMO, it might be best not to think of it as a value at all.
>>>>
>>>> Cantor called it a number.
>>>
>>> A number representing an order type.
>>
>> like 2 or 5.
>
> No, because 2 or 5 were FINITE numbers, not transfinite numbers.

Learn what an order type is according to Cantor.

Regards, WM

WM submitted this idea :
> Le 26/04/2024 à 01:03, FromTheRafters a écrit :
>> After serious thinking WM wrote :
>>> Le 24/04/2024 à 20:08, FromTheRafters a écrit :
>>>> WM explained :
>>>>> Le 24/04/2024 à 12:49, FromTheRafters a écrit :
>>>>>> Richard Damon wrote :
>>>>>
>>>>>>> Of course, if you logic tries to make that transfinite value finite
>>>>>>> because that is all it can handle, your system just blows up in your
>>>>>>> face.
>>>>>>
>>>>>> IMO, it might be best not to think of it as a value at all.
>>>>>
>>>>> Cantor called it a number.
>>>>
>>>> A number representing an order type.
>>>
>>> like 2 or 5.
>>
>> No, like:

The part you left out:

https://en.wikipedia.org/wiki/Order_type#Order_type_of_well-orderings

> Read Cantor. It is not difficult. Even you can understand it.
> Jeder geordneten Menge M kommt ein bestimmter "Ordnungstypus" oder kürzer ein
> bestimmter "Typus" zu.

Dishonest fuckhead!

On 4/26/24 1:02 PM, WM wrote:
> Le 26/04/2024 à 16:49, Richard Damon a écrit :
>> On 4/26/24 10:41 AM, WM wrote:
>
>>>
>>> The set is bounded by ω.
>>
>> But ω isn't in the set, so it can't be the upper bound of the set that
>> is below ω
>
> Learn the meaning of upper bound. ω*2 is an upper bound too.

But since it isn't in the set, you can't use it as the value to "step
back" to.

Note, that value isn't "of the set that is below ω".

>
> Fact is that below ω there are natnumbers and above there are none.
> Hence they cease below the bound ω.

Yes, but it can't be the "Upper Bound" used to step back to from ω.

There is not "Upper Bound" for the Natural Numbers *IN* the Natural
Numbers to be the "last" value to step back to.

Thus, your logic about ω-1 breaks. The value does not exist, as there
can not be a "transfinite" value below ω, by its definition, and ω-1
can't be a Natural Number, as then it would be the "last" Natural
Number, but that set doesn't have a "last" member, as its "Upper Bound"
is outside the set, as it is a "Unbounded" set.

>
> Regards, WM

Richard Damon schrieb:
> On 4/26/24 1:02 PM, WM wrote:
>> Le 26/04/2024 à 16:49, Richard Damon a écrit :
>>> On 4/26/24 10:41 AM, WM wrote:
>>
>>>>
>>>> The set is bounded by ω.
>>>
>>> But ω isn't in the set, so it can't be the upper bound of the set that
>>> is below ω
>>
>> Learn the meaning of upper bound. ω*2 is an upper bound too.
>
> But since it isn't in the set, you can't use it as the value to "step
> back" to.
>
> Note, that value isn't "of the set that is below ω".
>
>>
>> Fact is that below ω there are natnumbers and above there are none.
>> Hence they cease below the bound ω.
>
> Yes, but it can't be the "Upper Bound" used to step back to from ω.
>
> There is not "Upper Bound" for the Natural Numbers *IN* the Natural
> Numbers to be the "last" value to step back to.
>
> Thus, your logic about ω-1 breaks. The value does not exist, as there
> can not be a "transfinite" value below ω, by its definition, and ω-1
> can't be a Natural Number, as then it would be the "last" Natural
> Number, but that set doesn't have a "last" member, as its "Upper Bound"
> is outside the set, as it is a "Unbounded" set.

Right, but WM will tell you (too), that this is "wrong" and that in the
"real mathematics" there *IS* ω-1 that has to be accepted (all the same
as a first fraction and a first real number left to the 0) and this is
what WM is hard fighting for 40++ years. WM does neither accept whatever
different statements folks have told here (and elsewhere) nor whatever
is written down in the textbooks. WM despises them (as "matheologicians",
as folks who believe in illogical, unnatural nonsense which is criminal
to teach to pupils ("wards") in the education system).

On 4/26/2024 10:37 AM, WM wrote:
> Le 26/04/2024 à 01:11, Jim Burns a écrit :
>> On 4/25/2024 4:03 PM, WM wrote:

>>> If all smaller numbers are doubled,
>>> then there is no place for
>>> the doubled numbers below ω.
>>
>> If n is below ω
>> then n can be counted to from 0
>> then n⋅2 can be counted to from n
>
> That is true for definable numbers
> but not for the last numbers before ω.

If any number below n canNOT be counted to from 0
then n itself canNOT be counted to from 0

Thus,
each number which CAN be counted to from 0
is not above
any number which canNOT be counted to from 0

By definition,
ω is between
numbers which CAN be counted to from 0 and
numbers which canNOT be counted to from 0

Imagine being someone who denies that definition of ω

Because the following isn't a claim about ω
you (the denier) should still admit:
if n can be counted to from 0
then n⋅2 can be counted to from n
then n⋅2 can be counted to from 0 (through n)

If ω exists as defined,
then doubling never crosses ω
(from CAN to canNOT)

Even if ω doesn't exist as defined,
then doubling never crosses
_where ω would be if ω existed_
(from CAN to canNOT)

ω is NOT a simply.humongous.instance of
the numbers 0 1 2 3 ...
ω marks a boundary between domains with
different descriptions (CAN and canNOT).

Imagine being someone who denies that
ω marks that boundary.
With or without the marker,
the domains (CAN and canNOT) remain
the domains (CAN and canNOT).

On 4/26/24 1:06 PM, WM wrote:
> Le 26/04/2024 à 04:08, Richard Damon a écrit :
>> On 4/25/24 3:51 PM, WM wrote:
>>> Le 24/04/2024 à 20:08, FromTheRafters a écrit :
>>>> WM explained :
>>>>> Le 24/04/2024 à 12:49, FromTheRafters a écrit :
>>>>>> Richard Damon wrote :
>>>>>
>>>>>>> Of course, if you logic tries to make that transfinite value
>>>>>>> finite because that is all it can handle, your system just blows
>>>>>>> up in your face.
>>>>>>
>>>>>> IMO, it might be best not to think of it as a value at all.
>>>>>
>>>>> Cantor called it a number.
>>>>
>>>> A number representing an order type.
>>>
>>> like 2 or 5.
>>
>> No, because 2 or 5 were FINITE numbers, not transfinite numbers.
>
> Learn what an order type is according to Cantor.
>
> Regards, WM
>

Maybe you should. "Order Types" are not "Numbers" but properties of how
you order the set of numbers.

"Order Type" has nothing to do with what is being discussed.

Jim Burns schrieb:
> On 4/26/2024 10:37 AM, WM wrote:
>> Le 26/04/2024 à 01:11, Jim Burns a écrit :
>>> On 4/25/2024 4:03 PM, WM wrote:
>
>>>> If all smaller numbers are doubled,
>>>> then there is no place for
>>>> the doubled numbers below ω.
>>>
>>> If n is below ω
>>> then n can be counted to from 0
>>> then n⋅2 can be counted to from n
>>
>> That is true for definable numbers
>> but not for the last numbers before ω.
>
> If any number below n canNOT be counted to from 0
> then n itself canNOT be counted to from 0
>
> Thus,
> each number which CAN be counted to from 0
> is not above
> any number which canNOT be counted to from 0
>
> By definition,
> ω is between
> numbers which CAN be counted to from 0 and
> numbers which canNOT be counted to from 0
>
> Imagine being someone who denies that definition of ω
>
> Because the following isn't a claim about ω
> you (the denier) should still admit:
> if n can be counted to from 0
> then n*2 can be counted to from n
> then n*2 can be counted to from 0 (through n)
>
> If ω exists as defined,
> then doubling never crosses ω
> (from CAN to canNOT)
>
> Even if ω doesn't exist as defined,
> then doubling never crosses
> _where ω would be if ω existed_
> (from CAN to canNOT)
>
>
> ω is NOT a simply.humongous.instance of
> the numbers 0 1 2 3 ...
> ω marks a boundary between domains with
> different descriptions (CAN and canNOT).
>
> Imagine being someone who denies that
> ω marks that boundary.
> With or without the marker,
> the domains (CAN and canNOT) remain
> the domains (CAN and canNOT).

This is really well put!

Unfortunately, WM is not interested in our ideas of (our) math and
logic but in his own (mostly read up upon) ideas and his flexible
and willingly deformable "true logic" which a "normal" person can
even "feel" -- but even more is WM interested in (we) WHAT folks
CLAIM and STATE about (our) math, more than about that math itself.

On 4/26/2024 7:37 AM, WM wrote:
> Le 26/04/2024 à 01:11, Jim Burns a écrit :
>> On 4/25/2024 4:03 PM, WM wrote:
>
>>> If all smaller numbers are doubled,
>>> then there is no place for
>>> the doubled numbers below ω.
>>
>> If n is below ω
>> then n can be counted to from 0
>> then n⋅2 can be counted to from n
>
> That is true for definable numbers but not for the last numbers before ω.

There is no such thing as the last numbers _before_ ω...

>
>>>> I defined a map for which ω+3 ⟼ 2⋅3
>
> To my knowledge you did not. And if so, why did you?
>
> Regards, WM

On 4/25/2024 1:35 PM, Ross Finlayson wrote:
> On 04/25/2024 12:10 PM, Chris M. Thomasson wrote:
>> On 4/24/2024 7:55 PM, Ross Finlayson wrote:
>>> On 04/24/2024 06:16 PM, Moebius wrote:
>>>> Am 25.04.2024 um 02:49 schrieb Chris M. Thomasson:
>>>>
>>>>> Deeper down the rabbit hole. There is an infinity between 1 and
>>>>> 1.25...
>>>>
>>>> Yeah. The real numbers (and hence the complex numbers too) ARE quite
>>>> "deep".
>>>>
>>>> Indeed, the Mandelbrot fractal is a nice "depiction" of that fact.
>>>>
>>>
>>> https://www.youtube.com/watch?v=ZjZr7F_kJmw&list=PLb7rLSBiE7F5_h5sSsWDQmbNGsmm97Fy5&index=34
>>>
>>> -
>>>
>>> You know if you read a derivation of the complex numbers,
>>> where it gets to defining division, there's more than
>>> one branch than the usual principal branch, so it sort
>>> makes complex numbers a bit more complex.
>>>
>>> Yet, the roots of unity can also be considered a variety of ways,
>>> other than complex analysis or the usual Eulerian-Gaussian analysis,
>>> and, "roots of zero" is quite a deal, and the identity-dimension
>>> is even a thing (in mathematics).
>>>
>>>
>>
>> Fwiw, check this out... We can store data in the n-ary roots of complex
>> numbers:
>>
>> https://groups.google.com/g/comp.lang.c++/c/bB1wA4wvoFc/m/GdzmMd41AQAJ
>>
>> :^)
>
> It seems a sort of "arithmetic coding", or, as a
> sort of "addressing scheme".

Kind of... Well, it shows how data can be stored in the roots of a
complex number. The fun part is that data can be within the roots that
comprise escape time fractals, like the Mandelbrot and Julia sets. Fun
to me.

>
> Then as a sort of "arithmetization", as after an
> "algebraization", or about and around that as
> algebra, arithmetic, and geometry get separate
> treatments while all sitting together in the numbers
> and geometry their algebras, then that a "geometrization"
> would also have that besides encoding offsets of
> whatever interpolated sorts in the bits of the
> mantissa of the complex numbers' components,
> that also what's true in the geometry in the
> diagram that is the complex plane about R^2,
> would also be true about the things.
>
> For example, say sometimes the alphabet is sequential,
> and other times, it's like some finite ring that's
> co-prime its modulus so like the roots ring around
> the roots of unity, then the various sieves down
> into the alphabet would have the same order.
>
> Then it's two properties at once.

Interesting! Btw, have you tried to compile and run it on your end?

For people that do not have a C++ compiler handy, it can be run on an
online C++ compiler, here is just one of many:

https://i.ibb.co/vmfzv0M/image.png

There is an very interesting part wrt trying to get it into a state
where it has errors wrt loading up data. For instance, try to store:

ABCABCABCABC instead of my name CHRIS, it gets an error:

ORIGIN POINT:(-0.75,0.059999999999999998)
LOADED POINT:(0,0)

loaded:00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000046ABCABCABCABC
load_err_sum:5.3680514269578706e-14

***** DATA CORRUPTED!!! Shi%! *****
MnWOvMQcgO.o: /tmp/MnWOvMQcgO.cpp:290: int main(): Assertion `stored ==
loaded' failed.
Aborted

However, if you study that error, the data is still there, right at the end!

0046ABCABCABCABC

Strange!

On 4/25/2024 1:06 PM, WM wrote:
> Le 24/04/2024 à 22:07, "Chris M. Thomasson" a écrit :
>> On 4/24/2024 10:16 AM, WM wrote:
>>> Le 24/04/2024 à 04:01, Richard Damon a écrit :
>>>> On 4/23/24 3:34 PM, WM wrote:
>>>>> Le 23/04/2024 à 01:01, Richard Damon a écrit :
>>>>>> On 4/22/24 10:15 AM, WM wrote:
>>>>>
>>>>>>> The results cannot be compressed to the interval (0, ω) of the
>>>>>>> set { 1, 2, 3, ...}. This shows that new numbers are generated by
>>>>>>> multiplication.
>>>>>>>
>>>>>> Of course they can be compressed into the interval (0, ω), as
>>>>>> every finite number n < ω, when doubled results in a finite number
>>>>>> 2n which is also < ω.
>>>>>
>>>>> Try to map the closed interval [0, ω]*2 = [0, ω*2].
>>>>> If [0, ω) --> [0, ω) and ω*2 --> ω*2, then ω*2 is the only image
>>>>> point in (ω, ω*2]. Infinitely many points remain empty. Crippled
>>>>> mathematics. Ugly. Inacceptable.
>>>
>>>> Why?
>>>
>>> Continuity.
>>>>
>>>> [0, ω]*2 = { [0, w), ω } *2 = {[0, w), ω*2} since the Natural
>>>> numbers (what [0, ω) represents) are closed under multiplication.
>>>
>>> Impossible if (0, ω) is completely filled. Every 2n is larger than n.
>> [...]
>>
>> How can it be 100% completely filled when its unbounded?
>
> Ask Cantor or Bolzano. It can also be completely counted - according to
> Cantor.

How can you completely count an infinite unbounded set? You have some
issues WM... Deal with it. Your folly is not our problem... ;^o

> But don't ask modern set theorists. They are either too stupid or too
> dishonest to give the correct answer.
>
> Regards, WM
>
>

Chris M. Thomasson schrieb:
> On 4/25/2024 1:06 PM, WM wrote:
>> Le 24/04/2024 à 22:07, "Chris M. Thomasson" a écrit :
>>> On 4/24/2024 10:16 AM, WM wrote:
>>>> Le 24/04/2024 à 04:01, Richard Damon a écrit :
>>>>> On 4/23/24 3:34 PM, WM wrote:
>>>>>> Le 23/04/2024 à 01:01, Richard Damon a écrit :
>>>>>>> On 4/22/24 10:15 AM, WM wrote:
>>>>>>
>>>>>>>> The results cannot be compressed to the interval (0, ω) of the
>>>>>>>> set { 1, 2, 3, ...}. This shows that new numbers are generated by
>>>>>>>> multiplication.
>>>>>>>>
>>>>>>> Of course they can be compressed into the interval (0, ω), as
>>>>>>> every finite number n < ω, when doubled results in a finite number
>>>>>>> 2n which is also < ω.
>>>>>>
>>>>>> Try to map the closed interval [0, ω]*2 = [0, ω*2].
>>>>>> If [0, ω) --> [0, ω) and ω*2 --> ω*2, then ω*2 is the only image
>>>>>> point in (ω, ω*2]. Infinitely many points remain empty. Crippled
>>>>>> mathematics. Ugly. Inacceptable.
>>>>
>>>>> Why?
>>>>
>>>> Continuity.
>>>>>
>>>>> [0, ω]*2 = { [0, w), ω } *2 = {[0, w), ω*2} since the Natural
>>>>> numbers (what [0, ω) represents) are closed under multiplication.
>>>>
>>>> Impossible if (0, ω) is completely filled. Every 2n is larger than n.
>>> [...]
>>>
>>> How can it be 100% completely filled when its unbounded?
>>
>> Ask Cantor or Bolzano. It can also be completely counted - according to
>> Cantor.
>
> How can you completely count an infinite unbounded set? You have some
> issues WM... Deal with it. Your folly is not our problem... ;^o

WM takes himself to be a "finitist" - he denies that there is somewhere
in the universe any infinity "realized" (judging that math is happening
in the mental space), so he combats this idea and exacts "the existence"
of a first real number right to zero and a last finite number left to w
altogether with any idea of inifinite objects elsewhere in the universe.

On 4/26/24 5:48 PM, Tom Bola wrote:
> Chris M. Thomasson schrieb:
>> On 4/25/2024 1:06 PM, WM wrote:
>>> Le 24/04/2024 à 22:07, "Chris M. Thomasson" a écrit :
>>>> On 4/24/2024 10:16 AM, WM wrote:
>>>>> Le 24/04/2024 à 04:01, Richard Damon a écrit :
>>>>>> On 4/23/24 3:34 PM, WM wrote:
>>>>>>> Le 23/04/2024 à 01:01, Richard Damon a écrit :
>>>>>>>> On 4/22/24 10:15 AM, WM wrote:
>>>>>>>
>>>>>>>>> The results cannot be compressed to the interval (0, ω) of the
>>>>>>>>> set { 1, 2, 3, ...}. This shows that new numbers are generated by
>>>>>>>>> multiplication.
>>>>>>>>>
>>>>>>>> Of course they can be compressed into the interval (0, ω), as
>>>>>>>> every finite number n < ω, when doubled results in a finite number
>>>>>>>> 2n which is also < ω.
>>>>>>>
>>>>>>> Try to map the closed interval [0, ω]*2 = [0, ω*2].
>>>>>>> If [0, ω) --> [0, ω) and ω*2 --> ω*2, then ω*2 is the only image
>>>>>>> point in (ω, ω*2]. Infinitely many points remain empty. Crippled
>>>>>>> mathematics. Ugly. Inacceptable.
>>>>>
>>>>>> Why?
>>>>>
>>>>> Continuity.
>>>>>>
>>>>>> [0, ω]*2 = { [0, w), ω } *2 = {[0, w), ω*2} since the Natural
>>>>>> numbers (what [0, ω) represents) are closed under multiplication.
>>>>>
>>>>> Impossible if (0, ω) is completely filled. Every 2n is larger than n.
>>>> [...]
>>>>
>>>> How can it be 100% completely filled when its unbounded?
>>>
>>> Ask Cantor or Bolzano. It can also be completely counted - according to
>>> Cantor.
>>
>> How can you completely count an infinite unbounded set? You have some
>> issues WM... Deal with it. Your folly is not our problem... ;^o
>
> WM takes himself to be a "finitist" - he denies that there is somewhere
> in the universe any infinity "realized" (judging that math is happening
> in the mental space), so he combats this idea and exacts "the existence"
> of a first real number right to zero and a last finite number left to w
> altogether with any idea of inifinite objects elsewhere in the universe.
>

But then he wants to talk about the transfinite numbers, which just can
not exist in the finite domain he tries to live in.

He seems to accept that mathematics does generate "infinite" numbers,
but then wants them to still act like the finite numbers he understands,
and complains about how they seem to break all the rules, so everything
needs to be come "dark" to hide the problems.

Richard Damon schrieb:
> On 4/26/24 5:48 PM, Tom Bola wrote:
>> Chris M. Thomasson schrieb:
>>> On 4/25/2024 1:06 PM, WM wrote:
>>>> Le 24/04/2024 à 22:07, "Chris M. Thomasson" a écrit :
>>>>> On 4/24/2024 10:16 AM, WM wrote:
>>>>>> Le 24/04/2024 à 04:01, Richard Damon a écrit :
>>>>>>> On 4/23/24 3:34 PM, WM wrote:
>>>>>>>> Le 23/04/2024 à 01:01, Richard Damon a écrit :
>>>>>>>>> On 4/22/24 10:15 AM, WM wrote:
>>>>>>>>
>>>>>>>>>> The results cannot be compressed to the interval (0, ω) of the
>>>>>>>>>> set { 1, 2, 3, ...}. This shows that new numbers are generated by
>>>>>>>>>> multiplication.
>>>>>>>>>>
>>>>>>>>> Of course they can be compressed into the interval (0, ω), as
>>>>>>>>> every finite number n < ω, when doubled results in a finite number
>>>>>>>>> 2n which is also < ω.
>>>>>>>>
>>>>>>>> Try to map the closed interval [0, ω]*2 = [0, ω*2].
>>>>>>>> If [0, ω) --> [0, ω) and ω*2 --> ω*2, then ω*2 is the only image
>>>>>>>> point in (ω, ω*2]. Infinitely many points remain empty. Crippled
>>>>>>>> mathematics. Ugly. Inacceptable.
>>>>>>
>>>>>>> Why?
>>>>>>
>>>>>> Continuity.
>>>>>>>
>>>>>>> [0, ω]*2 = { [0, w), ω } *2 = {[0, w), ω*2} since the Natural
>>>>>>> numbers (what [0, ω) represents) are closed under multiplication.
>>>>>>
>>>>>> Impossible if (0, ω) is completely filled. Every 2n is larger than n.
>>>>> [...]
>>>>>
>>>>> How can it be 100% completely filled when its unbounded?
>>>>
>>>> Ask Cantor or Bolzano. It can also be completely counted - according to
>>>> Cantor.
>>>
>>> How can you completely count an infinite unbounded set? You have some
>>> issues WM... Deal with it. Your folly is not our problem... ;^o
>>
>> WM takes himself to be a "finitist" - he denies that there is somewhere
>> in the universe any infinity "realized" (judging that math is happening
>> in the mental space), so he combats this idea and exacts "the existence"
>> of a first real number right to zero and a last finite number left to w
>> altogether with any idea of inifinite objects elsewhere in the universe.
>>
>
> But then he wants to talk about the transfinite numbers, which just can
> not exist in the finite domain he tries to live in.

Yes, he wants to refuse their existence, because only finite objects exist.

> He seems to accept that mathematics does generate "infinite" numbers,

Sure - our wrong brains do ... however, so.

> but then wants them to still act like the finite numbers he understands,

Yep.
> and complains about how they seem to break all the rules,

WM is not a bomb of intelligence, but kind of dense ;)

> so everything needs to be come "dark" to hide the problems.

He wants to talk all day...

Le 26/04/2024 à 20:13, Richard Damon a écrit :
> On 4/26/24 1:02 PM, WM wrote:
>> Le 26/04/2024 à 16:49, Richard Damon a écrit :
>>> On 4/26/24 10:41 AM, WM wrote:
>>
>>>>
>>>> The set is bounded by ω.
>>>
>>> But ω isn't in the set, so it can't be the upper bound of the set that
>>> is below ω
>>
>> Learn the meaning of upper bound. ω*2 is an upper bound too.
>
> But since it isn't in the set, you can't use it as the value to "step
> back" to.

The question was whether it is an upper bound. Bounds need not belong to a
set.
>
> Note, that value isn't "of the set that is below ω".

Therefore this set ends before.
>
>> Fact is that below ω there are natnumbers and above there are none.
>> Hence they cease below the bound ω.
>
> Yes, but it can't be the "Upper Bound" used to step back to from ω.

Then something else must be used.
>
> There is not "Upper Bound" for the Natural Numbers *IN* the Natural
> Numbers to be the "last" value to step back to.
>
> Thus, your logic about ω-1 breaks. The value does not exist,

In actual infinity something is below ω.

Regards, WM