On 14.11.2023 01:17, Jim Burns wrote:
> On 11/13/2023 6:33 PM, Fritz Feldhase wrote:
>> On Monday, November 13, 2023
>> at 11:30:28 PM UTC+1, Jim Burns wrote:
>>> On 11/13/2023 2:24 PM, WM wrote:
>
>>>> If every definable natural number has
>>>> infinitely many natural successors,

> Yes.
> However,
> I think WM means something else.

There is no other meaning. ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
That means there are ℵo dark numbers following upon all visible numbers.
Otherwise there would be a visible number without ℵo dark numbers
following upon it.

Regards, WM

wtorek, 14 listopada 2023 o 13:04:21 UTC+1 WM napisał(a):
> On 14.11.2023 07:09, Adam Polak wrote:
> > poniedziałek, 13 listopada 2023 o 20:24:24 UTC+1 WM napisał(a):

> If you wish to call 0 now -2, and so on, then both sets are the same. If
> you use the names for the well-known numbers, then both sets differ,
> namely by two elements of the second set which are missing in the first set.
>
> Regards, WM

And don't wish anything. I'm just explaining.

Look, from a !quantitative perspective!
these sets:
(A) 1, 2, 3, 4, 5, 6, 7, 8, ...
(B) I, II, III, 4, 5, 6, 7, 8, ...
(C) &, P, !, 4, 5, 6, 7, 8, ...
are equivalent.
I hope this is understandable.

Note, changing the "names" (i.e. quality parameters) of three elements of the set did not bringed change the number of elements.

This exact rule is ALWAYS in force!
even when it comes to changing the "name" of each element in the set.

This set: { 1, 2, 3, 144 }
is equinumerous!, with this set: { 1, 2, 7, 144 }
and with this one: { 5, 6, 7, 144 }
and with this one: { 777, 1490, @, P, 12 }

When you understand that changing the "names", i.e. changing the quality parameters of objects, does not change the number of elements in the set in ANY case!
you will no longer be surprised that:
this set:
1, 2, 3, 4, 5, 6, 7, 8, ...
is equivalent to this one:
2, 4, 6, 8, 10, 12, 14, 16...
and with this one:
1, 2, 3, 4, 5, 6, 7, 8, ...
and with the one that "contains all" these elements:
1/1, 1/2, 1/3, ...
2/1, 2/2, 2/3, ...
3/1, 3/2, 3/3, ...
.... , ... , ... , ...

Simply, all sets containing n elements (for n belonging to N) are quantitatively equal.
Exactly like all infinite sets - interpreted as "containing oo number" of elements, they are quantitativeli equal.

BR,
Adam

WM expressed precisely :
> On 14.11.2023 01:17, Jim Burns wrote:
>> On 11/13/2023 6:33 PM, Fritz Feldhase wrote:
>>> On Monday, November 13, 2023
>>> at 11:30:28 PM UTC+1, Jim Burns wrote:
>>>> On 11/13/2023 2:24 PM, WM wrote:
>>
>>>>> If every definable natural number has
>>>>> infinitely many natural successors,
>
>> Yes.
>> However,
>> I think WM means something else.
>
> There is no other meaning. ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> That means there are ℵo dark numbers following upon all visible numbers.
> Otherwise there would be a visible number without ℵo dark numbers following
> upon it.

No it doesn't, it means that a difference set between the naturals and
a finite subset of the naturals has cardinality aleph_zero.

on 11/14/2023, Adam Polak supposed :
> wtorek, 14 listopada 2023 o 13:04:21 UTC+1 WM napisał(a):
>> On 14.11.2023 07:09, Adam Polak wrote:
>>> poniedziałek, 13 listopada 2023 o 20:24:24 UTC+1 WM napisał(a):
>
>> If you wish to call 0 now -2, and so on, then both sets are the same. If
>> you use the names for the well-known numbers, then both sets differ,
>> namely by two elements of the second set which are missing in the first set.
>>
>> Regards, WM
>
> And don't wish anything. I'm just explaining.
>
> Look, from a !quantitative perspective!
> these sets:
> (A) 1, 2, 3, 4, 5, 6, 7, 8, ...
> (B) I, II, III, 4, 5, 6, 7, 8, ...
> (C) &, P, !, 4, 5, 6, 7, 8, ...
> are equivalent.
> I hope this is understandable.
>
> Note, changing the "names" (i.e. quality parameters) of three elements of the
> set did not bringed change the number of elements.
>
> This exact rule is ALWAYS in force!
> even when it comes to changing the "name" of each element in the set.
>
> This set: { 1, 2, 3, 144 }
> is equinumerous!, with this set: { 1, 2, 7, 144 }
> and with this one: { 5, 6, 7, 144 }
> and with this one: { 777, 1490, @, P, 12 }

A set with four elements is equinumerous with one with five elements?

Maybe you should include a non-infinite emptyset with your system so
that arithmetic works?

On 14.11.2023 14:02, Adam Polak wrote:
> wtorek, 14 listopada 2023 o 13:04:21 UTC+1 WM napisał(a):
>> On 14.11.2023 07:09, Adam Polak wrote:
>>> poniedziałek, 13 listopada 2023 o 20:24:24 UTC+1 WM napisał(a):
>
>> If you wish to call 0 now -2, and so on, then both sets are the same. If
>> you use the names for the well-known numbers, then both sets differ,
>> namely by two elements of the second set which are missing in the first set.
>>
>
> And don't wish anything. I'm just explaining.

The you should see that {0, 1, 2, 3, 4, 5, 6, 7, ...} is in bijection
with itself and therefore {-2, -1, 0, 1, 2, 3, 4, 5, ..} has two more
elements. In actual infinity all elements are existing, independently of
who uses them in whatever constellation.

> Look, from a !quantitative perspective!
> these sets:
> (A) 1, 2, 3, 4, 5, 6, 7, 8, ...
> (B) I, II, III, 4, 5, 6, 7, 8, ...
> (C) &, P, !, 4, 5, 6, 7, 8, ...
> are equivalent
They are even equinumerous, because {4, 5, 6, 7, 8, ...} is in bijection
with itself and {1, 2, 3} and {I, II, III} and {&, P, !}, all have
cardinality 3.

>
> Note, changing the "names" (i.e. quality parameters) of three elements of the set did not bringed change the number of elements.

Of course. Cantor stated this already: "the cardinal number |M| remains
unchanged if in place of an element or of each of some elements, or even
of each of all elements m of M another thing is substituted." [E.
Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und
philosophischen Inhalts", Springer, Berlin (1932) p. 283]
>
> This exact rule is ALWAYS in force!
> even when it comes to changing the "name" of each element in the set.
>
> This set: { 1, 2, 3, 144 }
> is equinumerous!, with this set: { 1, 2, 7, 144 }
> and with this one: { 5, 6, 7, 144 }

Of course, we agree.

> and with this one: { 777, 1490, @, P, 12 }

Not quite.
>
> When you understand that changing the "names", i.e. changing the quality parameters of objects, does not change the number of elements in the set in ANY case!
> you will no longer be surprised that:
> this set:
> 1, 2, 3, 4, 5, 6, 7, 8, ...
> is equivalent to this one:
> 2, 4, 6, 8, 10, 12, 14, 16...

That is wrong because there are not enough even natural numbers that
could be substituted. There are only half as many even numbers as
natural numbers.

> and with the one that "contains all" these elements:
> 1/1, 1/2, 1/3, ...
> 2/1, 2/2, 2/3, ...
> 3/1, 3/2, 3/3, ...
> ... , ... , ... , ...

No, that is wrong! In every line there are |ℕ| elements. In every column
there are |ℕ| elements. The complete matrix contains |ℕ|^2 elements. For
proof that it cannot be enumerated see "A game like billiards"
https://groups.google.com/g/sci.math/c/lH60V21yyi4.
>
> Simply, all sets containing n elements (for n belonging to N) are quantitatively equal.
> Exactly like all infinite sets - interpreted as "containing oo number" of elements, they are quantitativeli equal.
>
oo denotes potential infinity. All such sets can be put in bijection.
But actual infinity forbids that.
|ℕ| - 1 =/= |ℕ| =/= |ℕ| + 1 and {0, 1, 2, 3, ..., ω} \ ℕ = {0, ω}.

Regards, WM

On 14.11.2023 15:30, FromTheRafters wrote:
> WM expressed precisely :

>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>> That means there are ℵo dark numbers following upon all visible
>> numbers. Otherwise there would be a visible number without ℵo dark
>> numbers following upon it.
>
> No it doesn't, it means that a difference set between the naturals and a
> finite subset of the naturals has cardinality aleph_zero.

Every definable initial segment of the naturals is finite. The union of
definable initial segments is potentially infinite, namely *always
finite*. Therefore the difference set is dark and actually infinite.

Try to find a natural number of the union of finite initial segments
which has not a set of ℵo dark numbers following it.

Regards, WM

On Tuesday, November 14, 2023 at 7:09:14 AM UTC+1, Adam Polak wrote:

> Regarding logic. Yes, it is crucial and is completely missing in set theory and <bla>

No, it's not "missing in set theory". Actually, the logical framework is CRUCIAL for modern/axiomatic set theory, idiot.

> 1, 2, 3, 4, 5, 6, 7, 8...
> I, II, III, IV, 5, 6, 7, 8, ...
> are [...] THE SAME SET[...]

It seems to me, that what you want to express is just: {1, 2, 3, 4, 5, 6, 7, 8...} = {I, II, III, IV, 5, 6, 7, 8, ...} .

This certainly is true, if the symbols "I", "II", "III", "IV" are used to refer to 1, 2, 3, 4, resp.

> 0, 1, 2, 3, 4, 5, 6, 7, ...
> is [...] the same set [...] as:
> -2, -1, 0, 1, 2, 3, 4, 5, ...

No, the set {0, 1, 2, 3, 4, 5, 6, 7, ...} is usually NOT considered identical with the set {-2, -1, 0, 1, 2, 3, 4, 5, ...}, since usually we assume that -2, -1 !e {0, 1, 2, 3, 4, 5, 6, 7, ... }.

> "all" "every"

(in the present context) refer to the /universal quantifier/. See: https://en.wikipedia.org/wiki/Universal_quantification

> 0.001
> and this:
> 0.0010000000000000...
> are two completely different objects

No, they aren't. Both expressions "0.001" and "0.0010000000000000..." refer to (denote) the very same /object/, namely the rational number 1/1000.

wtorek, 14 listopada 2023 o 15:35:58 UTC+1 FromTheRafters napisał(a):
> on 11/14/2023, Adam Polak supposed :
> > wtorek, 14 listopada 2023 o 13:04:21 UTC+1 WM napisał(a):
> >> On 14.11.2023 07:09, Adam Polak wrote:
> >>> poniedziałek, 13 listopada 2023 o 20:24:24 UTC+1 WM napisał(a):

> > This set: { 1, 2, 3, 144 }
> > is equinumerous!, with this set: { 1, 2, 7, 144 }
> > and with this one: { 5, 6, 7, 144 }
> > and with this one: { 777, 1490, @, P, 12 }
> A set with four elements is equinumerous with one with five elements?
>
> Maybe you should include a non-infinite emptyset with your system so
> that arithmetic works?

Of course, you're right, I inserted one element too many in this example: { 777, 1490, @, P, 12 }
It should be e.g.: { 777, 1490, @, P }
Thank you for remark

BR, Adam

On 11/14/2023 7:10 AM, WM wrote:
> On 14.11.2023 01:17, Jim Burns wrote:
>> On 11/13/2023 6:33 PM, Fritz Feldhase wrote:
>>> On Monday, November 13, 2023
>>> at 11:30:28 PM UTC+1, Jim Burns wrote:
>>>> On 11/13/2023 2:24 PM, WM wrote:

>>>>> If every definable natural number has
>>>>> infinitely many natural successors,
>>>>> then all definable natural numbers have
>>>>> infinitely many natural successors.

>>> If every x in A has the property P,
>>> then all x in A have the property P.

>> Yes.
>> However,
>> I think WM means something else.
>
> There is no other meaning.
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> That means there are ℵo dark numbers
> following upon all visible numbers.

∀n ∈ ℕ_def: |ℕ_def\{1,2,3,...,n}| = ℵ₀
That means that,
for each visible number,
there are ℵ₀ visible numbers following.
Which is true.

∀⟨1,…,n⟩ ⊆ ℕ_def
ℕ_def = ⋂{S⊆ℕ_def| ∀⟨1,…,n⟩ ⊆ S }

> Otherwise
> there would be a visible number without
> ℵo dark numbers following upon it.

There are only visible numbers in ℕ_def
Nevertheless,
∀n ∈ ℕ_def: |ℕ_def\{1,2,3,...,n}| = ℵ₀

Adam Polak used his keyboard to write :
> poniedziałek, 13 listopada 2023 o 20:24:24 UTC+1 WM napisał(a):
>
>> If you mean actual infinity, that is right. But potential infinity is
>> only a never ending sequence of finite sets.
>
>> Nevertheless even in actually infinite sets basic logic is valid.
>> Example: If every definable natural number has infinitely many natural
>> successors, then all definable natural numbers have infinitely many
>> natural successors.
>
> Regarding logic. Yes, it is crucial and is completely missing in set theory
> and elsewhere, unfortunately.
>
> "everyone" and "all" have no quantitative context when used to describe an
> infinite set (infinity), ONLY qualitative! If you interpret it differently,
> you are making a mistake. Look, these:
> 1, 2, 3, 4, 5, 6, 7, 8...
> I, II, III, IV, 5, 6, 7, 8, ...
> are QUANTITATIVELY
> EXACTLY THE SAME SET/ THE SAME CONSTRUCTION
> they differ only in the qualitative parameters of a few elements
>
> consequently, this:
> 0, 1, 2, 3, 4, 5, 6, 7, ...
> is also the same set (from QUANTITATIVE perspective!)
> same as:
> -2, -1, 0, 1, 2, 3, 4, 5, ...
> and so on
>
> It can be said that in a quantitative context there is ONLY ONE infinite set.
> Or to put it another way, there is only one model of the infinite set, with
> which - in a quantitative context! - EVERY infinite set, WITHOUT EXCEPTION,
> is consistent. The differences between infinite sets come down only to
> differences in the qualitative parameters of individual objects that are
> elements of different infinite sets.
>
> It is impossible to understand these issues if you do not understand a number
> of subtle but key issues, e.g.: as I wrote above: "all" "every" only in the
> context of qualitative parameters, never in the quantitative context in
> relation to an infinite set.
>
> Another extremely important issue: Number and numerical value
>
> "numerical value" - is not a mathematical object, it is only a qualitative
> parameter that an object may have,
>
> "a number" - is a mathematical object with a qualitative parameter that is
> numerical value, preferably if it is the only qualitative parameter of the
> number, if additional ones appear, problems will also arise e.g.:
> 4 - "four" (number, mathematical object)
> IV - "four" (number, mathematical object)
> but!:
> IV - "four Roman" (it is an object with an additional quality parameter
> "Roman", as a result it is no longer a correctly defined number, it is not "a
> number" anymore)
>
> This may seem like little or nothing important all, but it's an illusion.
> This exact error leads to the conclusion that there may be objects other than
> irrational numbers in the right column of the matrix (Cantor's Diagonal
> Method). This is simply not true. This: 0.001
> and this:
> 0.0010000000000000...
> are two completely different objects
> only the first of which is a properly constructed decimal, "a number".

I believe it is only by convention that we drop the trailing zeros in
the infinite continued decimal expansion representation and call it
terminated.

0.0009999... or 0.000(9) is another unending CDE for that same
mathematical object.

On Tuesday, November 14, 2023 at 10:21:57 PM UTC+1, Jim Burns wrote:
> On 11/14/2023 7:10 AM, WM wrote:
> >
> > ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
> >
> > That means there are ℵo dark numbers
> > following upon all visible numbers.

Errr...

What WM (this fucking asshole full of shit) keeps to ignore:

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

That means that for each and every natural number n the difference between IN and {1, ..., n} is countably infinte.

Equivalently: For each and every natural number n there are ℵ₀ natural numbers following (larger than) n.

_____________________________________________________________

This fucking asshole full of shit keep talking about "dark numbers" and "visible numbers", NO ONE ELSE is interested in.

środa, 15 listopada 2023 o 00:20:50 UTC+1 FromTheRafters napisał(a):
> Adam Polak used his keyboard to write :
> > poniedziałek, 13 listopada 2023 o 20:24:24 UTC+1 WM napisał(a):

> I believe it is only by convention that we drop the trailing zeros in
> the infinite continued decimal expansion representation and call it
> terminated.
>
> 0.0009999... or 0.000(9) is another unending CDE for that same
> mathematical object.

Yes, conventions, both good ones and those imperfect ones.

"Task for children" :) Find the difference between the pictures:

Pict No. 1 Pict No. 2

1/2 0.5

I'm sure everyone can spot it.
Each "picture" above shows a DIFFERENT object - these are two different mathematical objects! - the differences are noticeable at first glance. Both objects have the same qualitative feature, which is their numerical value.

They can be used interchangeably ONLY as long as only one of their properties, only one quality parameter of these objects (it's numerical value) is taken into account during applications, actions on/with these objects.
When another property is taken into account during activities, e.g., construction of object, e.g. meaning will be given to which of the objects has "/" as its element, these objects become completely different and independent..

This: 0.0009999... and this: 0.000(9)
it is the same only as long as you pretend not to notice the differences that are evident and completely ignore these differences in the actions on/with these objects.

This can not be puted in Cantor's Diagonal Method matrix:
0.001
isn't it?
and this one can:
0.0010000000000000...

Why? because it is determined by the structure of the object, not buy numerical value. Which means that in this case: 0.0010000000000000... is not correctly defined "the number" known as: "one thousandth".

Cantor's Diagonal Method matrix in it's well known form is flaweded.

"Task for children" :) Find the difference between the pictures:

Pict No. 1 Pict No. 2

1/2 0.5

środa, 15 listopada 2023 o 00:20:50 UTC+1 FromTheRafters napisał(a):
> Adam Polak used his keyboard to write :
> > poniedziałek, 13 listopada 2023 o 20:24:24 UTC+1 WM napisał(a):
> >

> I believe it is only by convention that we drop the trailing zeros in
> the infinite continued decimal expansion representation and call it
> terminated.
>
> 0.0009999... or 0.000(9) is another unending CDE for that same
> mathematical object.

Yes, conventions, both good ones and those imperfect ones.

"Task for children" :) Find the difference between the pictures:

Pict No. 1: 1/2

Pict No. 2: 0.5

I'm sure everyone can spot it.
Each "picture" above shows a DIFFERENT object - these are two different mathematical objects! - the differences are noticeable at first glance. Both objects have the same qualitative feature, which is their numerical value.

They can be used interchangeably ONLY as long as only one of their properties, only one quality parameter of these objects (it's numerical value) is taken into account during applications, actions on/with these objects.
When another property is taken into account during activities, e.g., construction of object, e.g. meaning will be given to which of the objects has "/" as its element, these objects become completely different and independent..

This: 0.0009999... and this: 0.000(9)
it is the same only as long as you pretend not to notice the differences that are evident and completely ignore these differences in the actions on/with these objects.

This can not be puted in Cantor's Diagonal Method matrix:
0.001
isn't it?
and this one can:
0.0010000000000000...

Why? because it is determined by the structure of the object, not buy numerical value. Which means that in this case: 0.0010000000000000... is not correctly defined "the number" known as: "one thousandth".

Cantor's Diagonal Method matrix in it's well known form is flaweded.

On 14.11.2023 22:21, Jim Burns wrote:
> On 11/14/2023 7:10 AM, WM wrote:

>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>> That means there are ℵo dark numbers
>> following upon all visible numbers.
>
> ∀n ∈ ℕ_def: |ℕ_def\{1,2,3,...,n}| = ℵ₀
> That means that,
> for each visible number,
> there are ℵ₀ visible numbers following.
> Which is true.

No that is wrong. ℕ_def is potentially infinite. There are oo visible
numbers following upon every visible number. But ℕ is complete. Best
this can be seen by the unit fractions.
The function NUF(x) cannot increase from NUF(0) = 0 to NUF(x>0) = ℵo.

You claim this antimathematical mantra of matheology. But it is refuted
by ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0, a finite distance.

Correct would be to say: For all x ∈ (0, 1] which are larger than at
least ℵo unit fractions and the gaps between them, NUF(x) = ℵo.
This cannot be all x > 0, because the unit fractions and the gaps
between them occupy points on the positive real axis.

>
> There are only visible numbers in ℕ_def
> Nevertheless,
> ∀n ∈ ℕ_def: |ℕ_def\{1,2,3,...,n}| = ℵ₀

Wrong. You will never have defined ℵ₀ natural numbers.

∀n ∈ ℕ_def: |ℕ_def\{1,2,3,...,n}| = oo.

∀n ∈ ℕ_def: Number of definable unit fractions between 0 and the
definable unit fraction 1/n = oo.

Regards, WM

Adam Polak used his keyboard to write :
> środa, 15 listopada 2023 o 00:20:50 UTC+1 FromTheRafters napisał(a):
>> Adam Polak used his keyboard to write :
>>> poniedziałek, 13 listopada 2023 o 20:24:24 UTC+1 WM napisał(a):
>
>> I believe it is only by convention that we drop the trailing zeros in
>> the infinite continued decimal expansion representation and call it
>> terminated.
>>
>> 0.0009999... or 0.000(9) is another unending CDE for that same
>> mathematical object.
>
> Yes, conventions, both good ones and those imperfect ones.
>
> "Task for children" :) Find the difference between the pictures:
>
> Pict No. 1 Pict No. 2
>
> 1/2 0.5
>
> I'm sure everyone can spot it.
> Each "picture" above shows a DIFFERENT object - these are two different
> mathematical objects! - the differences are noticeable at first glance. Both
> objects have the same qualitative feature, which is their numerical value.

I disagree. Mathematically speaking those are two different symbolic
representations of the same mathematical object.

[...]

środa, 15 listopada 2023 o 13:37:24 UTC+1 FromTheRafters napisał(a):
> Adam Polak used his keyboard to write :
> > środa, 15 listopada 2023 o 00:20:50 UTC+1 FromTheRafters napisał(a):
> >> Adam Polak used his keyboard to write :
> >>> poniedziałek, 13 listopada 2023 o 20:24:24 UTC+1 WM napisał(a):
> >
> >> I believe it is only by convention that we drop the trailing zeros in
> >> the infinite continued decimal expansion representation and call it
> >> terminated.
> >>
> >> 0.0009999... or 0.000(9) is another unending CDE for that same
> >> mathematical object.
> >
> > Yes, conventions, both good ones and those imperfect ones.
> >
> > "Task for children" :) Find the difference between the pictures:
> >
> > Pict No. 1 Pict No. 2
> >
> > 1/2 0.5
> >
> > I'm sure everyone can spot it.
> > Each "picture" above shows a DIFFERENT object - these are two different
> > mathematical objects! - the differences are noticeable at first glance. Both
> > objects have the same qualitative feature, which is their numerical value.
> I disagree. Mathematically speaking those are two different symbolic
> representations of the same mathematical object.
>
> [...]

You can, of course, disagree with reality, but you can't change it.
this: 1
this: 23/23
this: 1.00
this: 1.(0)
are completely different objects, which can be seen at first glance, having one common parameter: a numerical value.
Mathematically speaking, mathematics as a science honestly admits that it does not know what the object known as "the number ONE" is, and so on.
The common mistakes, I describe, are therefore not surprising. What is surprising that is the firmness of statements made in the absence of arguments,
because this: "Mathematically speaking" is not an argument at all.
Especially if you are aware that the truth and the official position of the scientific community is: mathematics as a science honestly admits that it does not know what the object known as "the number ONE" is.

BR,
Adam

After serious thinking Adam Polak wrote :
> środa, 15 listopada 2023 o 13:37:24 UTC+1 FromTheRafters napisał(a):
>> Adam Polak used his keyboard to write :
>>> środa, 15 listopada 2023 o 00:20:50 UTC+1 FromTheRafters napisał(a):
>>>> Adam Polak used his keyboard to write :
>>>>> poniedziałek, 13 listopada 2023 o 20:24:24 UTC+1 WM napisał(a):
>>>> I believe it is only by convention that we drop the trailing zeros in
>>>> the infinite continued decimal expansion representation and call it
>>>> terminated.
>>>>
>>>> 0.0009999... or 0.000(9) is another unending CDE for that same
>>>> mathematical object.
>>>
>>> Yes, conventions, both good ones and those imperfect ones.
>>>
>>> "Task for children" :) Find the difference between the pictures:
>>>
>>> Pict No. 1 Pict No. 2
>>>
>>> 1/2 0.5
>>>
>>> I'm sure everyone can spot it.
>>> Each "picture" above shows a DIFFERENT object - these are two different
>>> mathematical objects! - the differences are noticeable at first glance.
>>> Both objects have the same qualitative feature, which is their numerical
>>> value.
>> I disagree. Mathematically speaking those are two different symbolic
>> representations of the same mathematical object.
>>
>> [...]
>
> You can, of course, disagree with reality, but you can't change it.
> this: 1
> this: 23/23
> this: 1.00
> this: 1.(0)
> are completely different objects,

No, they are different symbols for the number one.

On Wednesday, November 15, 2023 at 3:21:19 PM UTC+1, FromTheRafters wrote:
> After serious thinking Adam Polak wrote :
> >
> > this: 1
> > this: 23/23
> > this: 1.00
> > this: 1.(0)
> > are completely different objects,

No, they aren't.

> No, they are different symbols for the number one.

Right.

1 = 23/23 = 1.00 = 1.(0)

Funny guy, our Adam.

On Wednesday, November 15, 2023 at 1:57:05 PM UTC+1, Adam Polak wrote:

> this: 1
> this: 23/23
> ...
> are completely different objects, which [...]

You are mixing up names/symbols/terms with the object(s) these names/symbols/terms refer to (denote).

The symbols "1", "23/23" both denote the number 1 (in a mathematical context like, say, analysis).

That's why we (may) write: 1 = 23/23.

On the other hand, clearly "1" =/= "23/23". (Hint: "1" consists of one character, while "23/23" consists of 5 characters.)

On Wednesday, November 15, 2023 at 10:11:00 AM UTC+1, Adam Polak wrote:
>
> Find the difference between the pictures:
>
> Pict No. 1: 1/2
>
> Pict No. 2: 0.5
>
>
> I'm sure everyone can spot it.

Sure.

What we SEE, are two different "symbolic expressions" (consisting of characters).

> the differences are noticeable at first glance.

Indeed! In "1/2" we spot the characters "1", "/", and "2". While in "0.5" we spot the characters "0", "." and "5".

> these are two different mathematical objects!

Nope. "1/2" and "0.5" are two different "symbols" (names, terms) which denote (in certain mathematical contexts like, say, analysis) one and the same mathematical object, namely the rational number 1/5.

Hence these symbols

> can be used interchangeably

(in certain mathematical contexts like, say, analysis).

Hint: 1/2 = 0.5 (there).

Same with

> This: 0.0009999... and this: 0.000(9)

Hint: 0.0009999... = 0.000(9) (there).

> <nonsense deleted>

on 11/15/2023, Fritz Feldhase supposed :
> On Wednesday, November 15, 2023 at 3:21:19 PM UTC+1, FromTheRafters wrote:
>> After serious thinking Adam Polak wrote :
>>>
>>> this: 1
>>> this: 23/23
>>> this: 1.00
>>> this: 1.(0)
>>> are completely different objects,
>
> No, they aren't.
>
>> No, they are different symbols for the number one.
>
> Right.
>
> 1 = 23/23 = 1.00 = 1.(0)
>
> Funny guy, our Adam.

I look forward to reading the arguments between WM and Adam Polak.

On Wednesday, November 15, 2023 at 7:11:51 PM UTC+1, FromTheRafters wrote:

> I look forward to reading the arguments between WM and Adam Polak.

Same, same. Crank fight! :-)

On 11/15/2023 6:54 AM, WM wrote:
> On 14.11.2023 22:21, Jim Burns wrote:
>> On 11/14/2023 7:10 AM, WM wrote:

>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>>> That means there are ℵo dark numbers
>>> following upon all visible numbers.
>>
>> ∀n ∈ ℕ_def: |ℕ_def\{1,2,3,...,n}| = ℵ₀
>> That means that,
>> for each visible number,
>> there are ℵ₀ visible numbers following.
>> Which is true.
>
> No that is wrong.
> ℕ_def is potentially infinite.

∀⟨1,…,n⟩ ⊆ ℕ_def
∀⟨1,…,n⟩ ⊆ A ⟹ A ⊇ ℕ_def

> There are oo visible numbers following upon
> every visible number.

For each n ∈ ℕ_def
there are |ℕ_def| numbers following upon n
in ℕ_def

For each n ∈ ℕ_def
n does not follow upon each k in ℕ_def

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

∀n ∈ ℕ_def:
~∀k ∈ ℕ_def: ℕ_def\{1,2,3,...,k} ∋ n

> But ℕ is complete.

Your completenessᵂᴹ requires things which
cannot be said to self-equal.
I pass.

>> There are only visible numbers in ℕ_def
>> Nevertheless,
>> ∀n ∈ ℕ_def: |ℕ_def\{1,2,3,...,n}| = ℵ₀
>
> Wrong.
> You will never have defined ℵ₀ natural numbers.

I have described ℕ_def
∀⟨1,…,n⟩ ⊆ ℕ_def
∀⟨1,…,n⟩ ⊆ A ⟹ A ⊇ ℕ_def

Starting from there,
we can augment not-first-false-ly.

On 11/15/2023 1:08 AM, Adam Polak wrote:
> "Task for children" :) Find the difference between the pictures:
>
> Pict No. 1 Pict No. 2
>
> 1/2 0.5

one half = one half

Find the difference:

..5 .25 * 2

1/2 (1/4)*2

You are a special kind of asshole.