An example is added about Cantor's set theory in the the section [Paradox Explanation]
https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download
and thought it may be interested:

+---------------------+
| Paradox Explanation |
+---------------------+
......
......
The number of even number and the number of natural number are equal: Ans:
ℕ=ℕ<0,+1> and ℕ<0,+2> are isomorphic but the "even numnber" in the two sets
are semantically different (i.e. the 6 in ℕ<0,+2> is 3 in ℕ<0,+1> ). They
are two different set of arithmetic systems. Thus, it is confusing to say
that the number of elements of an infinite set and its proper subset are
equal.

On 09/03/2024 11:45, wij wrote:
> An example is added about Cantor's set theory in the the section [Paradox Explanation]
> https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download
> and thought it may be interested:
>
> +---------------------+
> | Paradox Explanation |
> +---------------------+
> .....
> .....
> The number of even number and the number of natural number are equal: Ans:
> ℕ=ℕ<0,+1> and ℕ<0,+2> are isomorphic but the "even numnber" in the two sets
> are semantically different (i.e. the 6 in ℕ<0,+2> is 3 in ℕ<0,+1> ). They
> are two different set of arithmetic systems. Thus, it is confusing to say
> that the number of elements of an infinite set and its proper subset are
> equal.
>

I agree that the chosen wording above is likely to confuse particulaly non-mathematicians. That's
why when mathematicians talk about infinite sets, they are careful to /define/ the phrases they use
to describe them.

For example, they typically would not say "The number of even number and the number of natural
number are equal", because that would require them to have previously defined "the number of" for an
infinite set. More likely they say one of the following:

(a) There is a 1-1 correspondence between the even numbers and the natural numbers
[That is hardly "confusing" to anybody, when the correspondence is demonstrated!]

(b) The set of even numbers and the set of natural numbers "have the same cardinality"
[Where "have the same cardinality" is defined as there existing a
1-1 correspondence between the elements of the two sets, i.e. same as (a).]

(c) The set of even numbers and the set of natural numbers are "the same size"
[...having /defined/ "the same size" as meaning exactly the same as (a)]

This approach avoids ever directly referring to the "number" of elements in the set.

Alternatively, perhaps a concept of "cardinal number" has previously been defined, and it's been
shown that each set corresponds with a unique cardinal number, such that sets have the same
associated cardinal number exactly when (a) above applies. Then it would also be OK to say:

(c) The /cardinality/ of the set of even number equals the /cardinality/ of the
set of natural numbers.

Even then I don't think mathematicians would say "The /number/ of even number and the /number/ of
natural number are equal". That's just unnecessarily imprecise.

Perhaps the only people who would talk about the "number" of elements in an infinite set are
non-mathematicians (most of the population!) dabbling in the subject. Particularly journalists
explaining to the general public, and ignorant cranks trying to demonstrate some particular problem
with infinite sets (while typically misrepresenting the conventional mathematical standpoint)...

Perhaps all this can be classified as a "paradox" about Cantor's set theory, but not in the sense of
any problem with the theory. "Paradox" just in the sense of "unintuitive result when contrasted
with finite sets".

Regards,
Mike.

On Sat, 2024-03-09 at 17:30 +0000, Mike Terry wrote:
> On 09/03/2024 11:45, wij wrote:
> > An example is added about Cantor's set theory in the the section [Paradox Explanation]
> > https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download
> > and thought it may be interested:
> >
> > +---------------------+
> > > Paradox Explanation |
> > +---------------------+
> > .....
> > .....
> > The number of even number and the number of natural number are equal: Ans:
> >      ℕ=ℕ<0,+1> and ℕ<0,+2> are isomorphic but the "even numnber" in the two sets
> >      are semantically different (i.e. the 6 in ℕ<0,+2> is 3 in ℕ<0,+1> ). They
> >      are two different set of arithmetic systems. Thus, it is confusing to say
> >      that the number of elements of an infinite set and its proper subset are
> >      equal.
> >
>
> I agree that the chosen wording above is likely to confuse particulaly non-mathematicians.  That's
> why when mathematicians talk about infinite sets, they are careful to /define/ the phrases they use
> to describe them.
>
> For example, they typically would not say "The number of even number and the number of natural
> number are equal", because that would require them to have previously defined "the number of" for an
> infinite set.  More likely they say one of the following:
>
> (a) There is a 1-1 correspondence between the even numbers and the natural numbers
>      [That is hardly "confusing" to anybody, when the correspondence is demonstrated!]
>
> (b) The set of even numbers and the set of natural numbers "have the same cardinality"
>      [Where "have the same cardinality" is defined as there existing a
>      1-1 correspondence between the elements of the two sets, i.e. same as (a).]
>
> (c) The set of even numbers and the set of natural numbers are "the same size"
>      [...having /defined/ "the same size" as meaning exactly the same as (a)]
>
> This approach avoids ever directly referring to the "number" of elements in the set.
>
> Alternatively, perhaps a concept of "cardinal number" has previously been defined, and it's been
> shown that each set corresponds with a unique cardinal number, such that sets have the same
> associated cardinal number exactly when (a) above applies.  Then it would also be OK to say:
>
> (c) The /cardinality/ of the set of even number equals the /cardinality/ of the
>      set of natural numbers.
>
> Even then I don't think mathematicians would say "The /number/ of even number and the /number/ of
> natural number are equal".  That's just unnecessarily imprecise.
>
> Perhaps the only people who would talk about the "number" of elements in an infinite set are
> non-mathematicians (most of the population!) dabbling in the subject.  Particularly journalists
> explaining to the general public, and ignorant cranks trying to demonstrate some particular problem
> with infinite sets (while typically misrepresenting the conventional mathematical standpoint)...
>
> Perhaps all this can be classified as a "paradox" about Cantor's set theory, but not in the sense of
> any problem with the theory.  "Paradox" just in the sense of "unintuitive result when contrasted
> with finite sets".
>
>
> Regards,
> Mike.
>

Stupid is everywhere. Every one can be stupid, every one can be olcott.
I interpret the response as a perfect demonstration of what the imaginary, stupid
mathematician would do: Using (lots) more confusing words to cover the fact or ignorance.

In the example N<0,+2>, where 6 is actually an odd number. So, what does the
'even number' mean? Does it refer to the 'real subset' of the set itself or
another set? Please provide a clear example that explains what you say in no
confusing way !

On 09.03.2024 18:30, Mike Terry wrote:

>
> I agree that the chosen wording above is likely to confuse particulaly
> non-mathematicians.  That's why when mathematicians talk about infinite
> sets, they are careful to /define/ the phrases they use to describe them.

And they will not accept that their definition is misleading.
>
> For example, they typically would not say "The number of even number and
> the number of natural number are equal", because that would require them
> to have previously defined "the number of" for an infinite set.  More
> likely they say one of the following:
>
> (a) There is a 1-1 correspondence between the even numbers and the
> natural numbers
>     [That is hardly "confusing" to anybody, when the correspondence is
> demonstrated!]
>
> (b) The set of even numbers and the set of natural numbers "have the
> same cardinality"
>     [Where "have the same cardinality" is defined as there existing a
>     1-1 correspondence between the elements of the two sets, i.e. same
> as (a).]
>
> (c) The set of even numbers and the set of natural numbers are "the same
> size"
>     [...having /defined/ "the same size" as meaning exactly the same as
> (a)]

The existence of infinite bijections has ben disproved:

All positive fractions

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
...

can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 + m
which attaches the index k to the fraction m/n in Cantor's sequence

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

Its terms can be represented by matrices. When we attach all indeXes k =
1, 2, 3, ..., for clarity represented by X, to the integer fractions m/1
and indicate missing indexes by hOles O, then we get the matrix M(0) as
starting position:

XOOO... XXOO... XXOO... XXXO... ... XXXX...
XOOO... OOOO... XOOO... XOOO... ... XXXX...
XOOO... XOOO... OOOO... OOOO... ... XXXX...
XOOO... XOOO... XOOO... OOOO... ... XXXX...
.... ... ... ... ...
M(0) M(2) M(3) M(4) M(∞)

M(1) is the same as M(0) because index 1 remains at 1/1. In M(2) index 2
from 2/1 has been attached to 1/2. In M(3) index 3 from 3/1 has been
attached to 2/1. In M(4) index 4 from 4/1 has been attached to 1/3.
Successively all fractions of the sequence get indexed. In the limit,
denoted by M(∞), we see no fraction without index remaining. Note that
the only difference to Cantor's enumeration is that Cantor does not
render account for the source of the indices.

Every X, representing the index k, when taken from its present fraction
m/n, is replaced by the O taken from the fraction to be indexed by this
k. Its last carrier m/n will be indexed later by another index.
Important is that, when continuing, no O can leave the matrix as long as
any index X blocks the only possible drain, i.e., the first column. And
if leaving, where should it settle?

As long as indexes are in the drain, no O has left. The presence of all
O indicates that almost all fractions are not indexed. And after all
indexes have been issued and the drain has become free, no indexes are
available which could index the remaining matrix elements, yet covered by O.

It should go without saying that by rearranging the X of M(0) never a
complete covering can be realized. Lossless transpositions cannot suffer
losses. The limit matrix M(∞) only shows what should have happened when
all fractions were indexed. Logic proves that this cannot have happened
by exchanges. The only explanation for finally seeing M(∞) is that there
are invisible matrix positions, existing already at the start. Obviously
by exchanging O and X no O can leave the matrix, but the O can disappear
by moving without end, from visible to invisible positions.

Regards, WM

On 2024-03-10 11:16:28 +0000, WM said:

> On 09.03.2024 18:30, Mike Terry wrote:
>
>>
>> I agree that the chosen wording above is likely to confuse particulaly
>> non-mathematicians.  That's why when mathematicians talk about infinite
>> sets, they are careful to /define/ the phrases they use to describe
>> them.
>
> And they will not accept that their definition is misleading.

In mathematics that is not relevant.

--
Mikko

On 10/03/2024 01:47, wij wrote:
> On Sat, 2024-03-09 at 17:30 +0000, Mike Terry wrote:
>> On 09/03/2024 11:45, wij wrote:
>>> An example is added about Cantor's set theory in the the section [Paradox Explanation]
>>> https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download
>>> and thought it may be interested:
>>>
>>> +---------------------+
>>>> Paradox Explanation |
>>> +---------------------+
>>> .....
>>> .....
>>> The number of even number and the number of natural number are equal: Ans:
>>>      ℕ=ℕ<0,+1> and ℕ<0,+2> are isomorphic but the "even numnber" in the two sets
>>>      are semantically different (i.e. the 6 in ℕ<0,+2> is 3 in ℕ<0,+1> ). They
>>>      are two different set of arithmetic systems. Thus, it is confusing to say
>>>      that the number of elements of an infinite set and its proper subset are
>>>      equal.
>>>
>>
>> I agree that the chosen wording above is likely to confuse particulaly non-mathematicians.  That's
>> why when mathematicians talk about infinite sets, they are careful to /define/ the phrases they use
>> to describe them.
>>
>> For example, they typically would not say "The number of even number and the number of natural
>> number are equal", because that would require them to have previously defined "the number of" for an
>> infinite set.  More likely they say one of the following:
>>
>> (a) There is a 1-1 correspondence between the even numbers and the natural numbers
>>      [That is hardly "confusing" to anybody, when the correspondence is demonstrated!]
>>
>> (b) The set of even numbers and the set of natural numbers "have the same cardinality"
>>      [Where "have the same cardinality" is defined as there existing a
>>      1-1 correspondence between the elements of the two sets, i.e. same as (a).]
>>
>> (c) The set of even numbers and the set of natural numbers are "the same size"
>>      [...having /defined/ "the same size" as meaning exactly the same as (a)]
>>
>> This approach avoids ever directly referring to the "number" of elements in the set.
>>
>> Alternatively, perhaps a concept of "cardinal number" has previously been defined, and it's been
>> shown that each set corresponds with a unique cardinal number, such that sets have the same
>> associated cardinal number exactly when (a) above applies.  Then it would also be OK to say:
>>
>> (c) The /cardinality/ of the set of even number equals the /cardinality/ of the
>>      set of natural numbers.
>>
>> Even then I don't think mathematicians would say "The /number/ of even number and the /number/ of
>> natural number are equal".  That's just unnecessarily imprecise.
>>
>> Perhaps the only people who would talk about the "number" of elements in an infinite set are
>> non-mathematicians (most of the population!) dabbling in the subject.  Particularly journalists
>> explaining to the general public, and ignorant cranks trying to demonstrate some particular problem
>> with infinite sets (while typically misrepresenting the conventional mathematical standpoint)...
>>
>> Perhaps all this can be classified as a "paradox" about Cantor's set theory, but not in the sense of
>> any problem with the theory.  "Paradox" just in the sense of "unintuitive result when contrasted
>> with finite sets".
>>
>>
>> Regards,
>> Mike.
>>
>
> Stupid is everywhere. Every one can be stupid, every one can be olcott.
> I interpret the response as a perfect demonstration of what the imaginary, stupid
> mathematician would do: Using (lots) more confusing words to cover the fact or ignorance.
>

If you didn't understand any term I used, just ask about it and I'll explain further. To be honest,
I didn't twig that you were the author of the quotation, or that you wanted any explanation for the
even/odd issue... (if I had, I'd have replied a bit differently)

> In the example N<0,+2>, where 6 is actually an odd number. So, what does the
> 'even number' mean? Does it refer to the 'real subset' of the set itself or
> another set? Please provide a clear example that explains what you say in no
> confusing way !

The quote is about comparing the sizes of sets, right? So we have two sets:

S1 = {1, 2, 3, 4, 5, ...}
S2 = {2, 4, 6, 8, 10, ...}

When Cantor/set theory says they are the same size, that is saying that there is a 1-1
correspondence [one-to-one correspondence] between the elements of the set. Maybe you didn't
understand what that is. It's a pairing of the elements of the sets, e.g. like this:

1 <----> 2
2 <----> 4
3 <----> 6
4 <----> 8
5 <----> 10
...
1-1 correspondence means every element of S1 appears on the left (above) exactly once, every element
of S2 appears on the right exactly once, so each element of S1 has a corresponding element in S2 and
vice versa. I think you would agree that the above does indeed demonstrate such a correspondence.

You are asking something about the 3 <----> 6 line, saying that 3 is "an odd number" and 6 is "even
if considered as the natural number 6, but odd in the sense that it is the 3rd entry in S2 and 3 is
odd". Or something like that. So for you, the "semantics" of 3 and 6 as individuals is different,
so there is some problem with the 1-1 correspondance, which confuses you...

My response is that we cannot call 3 or 6 even or odd without a lot more "structure" than just the
bare sets S1 and S2: as a minimum we need to take into account the addition operations which are
separate structures from S1 and S2 themselves. And the key point is that when we are comparing the
sizes of two sets, WE DISREGARD ALL THAT "EXTRA STRUCTURE" stuff (what I think you refer to as the
"semantics" of the elements. We focus just on the individual elements themselves, as though they
are simply "distinct individuals" in some sense, which are simply to be paired with elements in the
other set.

For this pairing process it is of no consequence /what the elements mean/. Just that they are
correctly paired together. (Or that such a pairing is not possible.)

Like when we have two sets
A = {1, 2, 3}
B = {A, B, C}
we can match the elements together:
1 <----> A
2 <----> B
3 <----> C
showing that in Cantor world the sets are the same size.

But then someone comes along and points out "the elements of A are numbers, while the elements of B
are letters! They have different semantics, so it is confusing to say the sets have the same size!"

Hopefully you see the point I'm trying to explain - the "semantics" of the elements is completely
irrelevant for the purposes of the pairing process, i.e. when we are comparing the "sizes" of the sets.

If that person /insists/ that they are still confused due to the different semantics of the
elements, what could be said to cheer them up? Only "just don't be confused! just check out the
correspondence - does it work or not?"

I'm not sure if this is really what's confusing you, or is it something else? Since the original
quote was discussing the "paradox" of a proper subset of N having "the same size" as N itself,
that's what I've focussed on explaining. I have not really explained about even/odd numbers,
because that question is nothing whatsoever to do with the "paradox" being discussed. (I'd be happy
to have a go explaining that too, if you want...)

Mike.

On Sun, 2024-03-10 at 17:38 +0000, Mike Terry wrote:
> On 10/03/2024 01:47, wij wrote:
> > On Sat, 2024-03-09 at 17:30 +0000, Mike Terry wrote:
> > > On 09/03/2024 11:45, wij wrote:
> > > > An example is added about Cantor's set theory in the the section [Paradox Explanation]
> > > > https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download
> > > > and thought it may be interested:
> > > >
> > > > +---------------------+
> > > > > Paradox Explanation |
> > > > +---------------------+
> > > > .....
> > > > .....
> > > > The number of even number and the number of natural number are equal: Ans:
> > > >       ℕ=ℕ<0,+1> and ℕ<0,+2> are isomorphic but the "even numnber" in the two sets
> > > >       are semantically different (i.e. the 6 in ℕ<0,+2> is 3 in ℕ<0,+1> ). They
> > > >       are two different set of arithmetic systems. Thus, it is confusing to say
> > > >       that the number of elements of an infinite set and its proper subset are
> > > >       equal.
> > > >
> > >
> > > I agree that the chosen wording above is likely to confuse particulaly non-mathematicians. 
> > > That's
> > > why when mathematicians talk about infinite sets, they are careful to /define/ the phrases they
> > > use
> > > to describe them.
> > >
> > > For example, they typically would not say "The number of even number and the number of natural
> > > number are equal", because that would require them to have previously defined "the number of"
> > > for an
> > > infinite set.  More likely they say one of the following:
> > >
> > > (a) There is a 1-1 correspondence between the even numbers and the natural numbers
> > >       [That is hardly "confusing" to anybody, when the correspondence is demonstrated!]
> > >
> > > (b) The set of even numbers and the set of natural numbers "have the same cardinality"
> > >       [Where "have the same cardinality" is defined as there existing a
> > >       1-1 correspondence between the elements of the two sets, i.e. same as (a).]
> > >
> > > (c) The set of even numbers and the set of natural numbers are "the same size"
> > >       [...having /defined/ "the same size" as meaning exactly the same as (a)]
> > >
> > > This approach avoids ever directly referring to the "number" of elements in the set.
> > >
> > > Alternatively, perhaps a concept of "cardinal number" has previously been defined, and it's been
> > > shown that each set corresponds with a unique cardinal number, such that sets have the same
> > > associated cardinal number exactly when (a) above applies.  Then it would also be OK to say:
> > >
> > > (c) The /cardinality/ of the set of even number equals the /cardinality/ of the
> > >       set of natural numbers.
> > >
> > > Even then I don't think mathematicians would say "The /number/ of even number and the /number/
> > > of
> > > natural number are equal".  That's just unnecessarily imprecise.
> > >
> > > Perhaps the only people who would talk about the "number" of elements in an infinite set are
> > > non-mathematicians (most of the population!) dabbling in the subject.  Particularly journalists
> > > explaining to the general public, and ignorant cranks trying to demonstrate some particular
> > > problem
> > > with infinite sets (while typically misrepresenting the conventional mathematical standpoint)...
> > >
> > > Perhaps all this can be classified as a "paradox" about Cantor's set theory, but not in the
> > > sense of
> > > any problem with the theory.  "Paradox" just in the sense of "unintuitive result when contrasted
> > > with finite sets".
> > >
> > >
> > > Regards,
> > > Mike.
> > >
> >
> > Stupid is everywhere. Every one can be stupid, every one can be olcott.
> > I interpret the response as a perfect demonstration of what the imaginary, stupid
> > mathematician would do: Using (lots) more confusing words to cover the fact or ignorance.
> >
>
> If you didn't understand any term I used, just ask about it and I'll explain further.  To be honest,
> I didn't twig that you were the author of the quotation, or that you wanted any explanation for the
> even/odd issue...  (if I had, I'd have replied a bit differently)
>
> > In the example N<0,+2>, where 6 is actually an odd number. So, what does the
> > 'even number' mean? Does it refer to the 'real subset' of the set itself or
> > another set? Please provide a clear example that explains what you say in no
> > confusing way !
>
> The quote is about comparing the sizes of sets, right?  So we have two sets:
>
>    S1 = {1, 2, 3, 4, 5, ...}
>    S2 = {2, 4, 6, 8, 10, ...}
>
> When Cantor/set theory says they are the same size, that is saying that there is a 1-1
> correspondence [one-to-one correspondence] between the elements of the set.  Maybe you didn't
> understand what that is.  It's a pairing of the elements of the sets, e.g. like this:
>
>    1  <---->  2
>    2  <---->  4
>    3  <---->  6
>    4  <---->  8
>    5  <---->  10
>    ...
> 1-1 correspondence means every element of S1 appears on the left (above) exactly once, every element
> of S2 appears on the right exactly once, so each element of S1 has a corresponding element in S2 and
> vice versa.  I think you would agree that the above does indeed demonstrate such a correspondence.
>
> You are asking something about the 3 <----> 6 line, saying that 3 is "an odd number" and 6 is "even
> if considered as the natural number 6, but odd in the sense that it is the 3rd entry in S2 and 3 is
> odd".  Or something like that.  So for you, the "semantics" of 3 and 6 as individuals is different,
> so there is some problem with the 1-1 correspondance, which confuses you....
>
> My response is that we cannot call 3 or 6 even or odd without a lot more "structure" than just the
> bare sets S1 and S2:  as a minimum we need to take into account the addition operations which are
> separate structures from S1 and S2 themselves.  And the key point is that when we are comparing the
> sizes of two sets, WE DISREGARD ALL THAT "EXTRA STRUCTURE" stuff (what I think you refer to as the
> "semantics" of the elements.  We focus just on the individual elements themselves, as though they
> are simply "distinct individuals" in some sense, which are simply to be paired with elements in the
> other set.
>
> For this pairing process it is of no consequence /what the elements mean/..  Just that they are
> correctly paired together.  (Or that such a pairing is not possible.)
>
> Like when we have two sets
>      A = {1, 2, 3}
>      B = {A, B, C}
> we can match the elements together:
>      1  <---->  A
>      2  <---->  B
>      3  <---->  C
> showing that in Cantor world the sets are the same size.
>
> But then someone comes along and points out "the elements of A are numbers, while the elements of B
> are letters!  They have different semantics, so it is confusing to say the sets have the same size!"
>
> Hopefully you see the point I'm trying to explain - the "semantics" of the elements is completely
> irrelevant for the purposes of the pairing process, i.e. when we are comparing the "sizes" of the
> sets.
>
> If that person /insists/ that they are still confused due to the different semantics of the
> elements, what could be said to cheer them up?  Only "just don't be confused! just check out the
> correspondence - does it work or not?"
>
> I'm not sure if this is really what's confusing you, or is it something else?  Since the original
> quote was discussing the "paradox" of a proper subset of N having "the same size" as N itself,
> that's what I've focussed on explaining.  I have not really explained about even/odd numbers,
> because that question is nothing whatsoever to do with the "paradox" being discussed.  (I'd be happy
> to have a go explaining that too, if you want...)
>
>
> Mike.
>

On Mon, 2024-03-11 at 12:18 +0800, wij wrote:
> On Sun, 2024-03-10 at 17:38 +0000, Mike Terry wrote:
> > On 10/03/2024 01:47, wij wrote:
> > > On Sat, 2024-03-09 at 17:30 +0000, Mike Terry wrote:
> > > > On 09/03/2024 11:45, wij wrote:
> > > > > An example is added about Cantor's set theory in the the section [Paradox Explanation]
> > > > > https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download
> > > > > and thought it may be interested:
> > > > >
> > > > > +---------------------+
> > > > > > Paradox Explanation |
> > > > > +---------------------+
> > > > > .....
> > > > > .....
> > > > > The number of even number and the number of natural number are equal: Ans:
> > > > >       ℕ=ℕ<0,+1> and ℕ<0,+2> are isomorphic but the "even numnber" in the two sets
> > > > >       are semantically different (i.e. the 6 in ℕ<0,+2> is 3 in ℕ<0,+1> ). They
> > > > >       are two different set of arithmetic systems. Thus, it is confusing to say
> > > > >       that the number of elements of an infinite set and its proper subset are
> > > > >       equal.
> > > > >
> > > >
> > > > I agree that the chosen wording above is likely to confuse particulaly non-mathematicians. 
> > > > That's
> > > > why when mathematicians talk about infinite sets, they are careful to /define/ the phrases
> > > > they
> > > > use
> > > > to describe them.
> > > >
> > > > For example, they typically would not say "The number of even number and the number of natural
> > > > number are equal", because that would require them to have previously defined "the number of"
> > > > for an
> > > > infinite set.  More likely they say one of the following:
> > > >
> > > > (a) There is a 1-1 correspondence between the even numbers and the natural numbers
> > > >       [That is hardly "confusing" to anybody, when the correspondence is demonstrated!]
> > > >
> > > > (b) The set of even numbers and the set of natural numbers "have the same cardinality"
> > > >       [Where "have the same cardinality" is defined as there existing a
> > > >       1-1 correspondence between the elements of the two sets, i.e. same as (a).]
> > > >
> > > > (c) The set of even numbers and the set of natural numbers are "the same size"
> > > >       [...having /defined/ "the same size" as meaning exactly the same as (a)]
> > > >
> > > > This approach avoids ever directly referring to the "number" of elements in the set.
> > > >
> > > > Alternatively, perhaps a concept of "cardinal number" has previously been defined, and it's
> > > > been
> > > > shown that each set corresponds with a unique cardinal number, such that sets have the same
> > > > associated cardinal number exactly when (a) above applies.  Then it would also be OK to say:
> > > >
> > > > (c) The /cardinality/ of the set of even number equals the /cardinality/ of the
> > > >       set of natural numbers.
> > > >
> > > > Even then I don't think mathematicians would say "The /number/ of even number and the /number/
> > > > of
> > > > natural number are equal".  That's just unnecessarily imprecise.
> > > >
> > > > Perhaps the only people who would talk about the "number" of elements in an infinite set are
> > > > non-mathematicians (most of the population!) dabbling in the subject.  Particularly
> > > > journalists
> > > > explaining to the general public, and ignorant cranks trying to demonstrate some particular
> > > > problem
> > > > with infinite sets (while typically misrepresenting the conventional mathematical
> > > > standpoint)...
> > > >
> > > > Perhaps all this can be classified as a "paradox" about Cantor's set theory, but not in the
> > > > sense of
> > > > any problem with the theory.  "Paradox" just in the sense of "unintuitive result when
> > > > contrasted
> > > > with finite sets".
> > > >
> > > >
> > > > Regards,
> > > > Mike.
> > > >
> > >
> > > Stupid is everywhere. Every one can be stupid, every one can be olcott.
> > > I interpret the response as a perfect demonstration of what the imaginary, stupid
> > > mathematician would do: Using (lots) more confusing words to cover the fact or ignorance.
> > >
> >
> > If you didn't understand any term I used, just ask about it and I'll explain further.  To be
> > honest,
> > I didn't twig that you were the author of the quotation, or that you wanted any explanation for
> > the
> > even/odd issue...  (if I had, I'd have replied a bit differently)
> >
> > > In the example N<0,+2>, where 6 is actually an odd number. So, what does the
> > > 'even number' mean? Does it refer to the 'real subset' of the set itself or
> > > another set? Please provide a clear example that explains what you say in no
> > > confusing way !
> >
> > The quote is about comparing the sizes of sets, right?  So we have two sets:
> >
> >    S1 = {1, 2, 3, 4, 5, ...}
> >    S2 = {2, 4, 6, 8, 10, ...}
> >
> > When Cantor/set theory says they are the same size, that is saying that there is a 1-1
> > correspondence [one-to-one correspondence] between the elements of the set.  Maybe you didn't
> > understand what that is.  It's a pairing of the elements of the sets, e.g. like this:
> >
> >    1  <---->  2
> >    2  <---->  4
> >    3  <---->  6
> >    4  <---->  8
> >    5  <---->  10
> >    ...
> > 1-1 correspondence means every element of S1 appears on the left (above) exactly once, every
> > element
> > of S2 appears on the right exactly once, so each element of S1 has a corresponding element in S2
> > and
> > vice versa.  I think you would agree that the above does indeed demonstrate such a correspondence.
> >
> > You are asking something about the 3 <----> 6 line, saying that 3 is "an odd number" and 6 is
> > "even
> > if considered as the natural number 6, but odd in the sense that it is the 3rd entry in S2 and 3
> > is
> > odd".  Or something like that.  So for you, the "semantics" of 3 and 6 as individuals is
> > different,
> > so there is some problem with the 1-1 correspondance, which confuses you...
> >
> > My response is that we cannot call 3 or 6 even or odd without a lot more "structure" than just the
> > bare sets S1 and S2:  as a minimum we need to take into account the addition operations which are
> > separate structures from S1 and S2 themselves.  And the key point is that when we are comparing
> > the
> > sizes of two sets, WE DISREGARD ALL THAT "EXTRA STRUCTURE" stuff (what I think you refer to as the
> > "semantics" of the elements.  We focus just on the individual elements themselves, as though they
> > are simply "distinct individuals" in some sense, which are simply to be paired with elements in
> > the
> > other set.
> >
> > For this pairing process it is of no consequence /what the elements mean/.  Just that they are
> > correctly paired together.  (Or that such a pairing is not possible.)
> >
> > Like when we have two sets
> >      A = {1, 2, 3}
> >      B = {A, B, C}
> > we can match the elements together:
> >      1  <---->  A
> >      2  <---->  B
> >      3  <---->  C
> > showing that in Cantor world the sets are the same size.
> >
> > But then someone comes along and points out "the elements of A are numbers, while the elements of
> > B
> > are letters!  They have different semantics, so it is confusing to say the sets have the same
> > size!"
> >
> > Hopefully you see the point I'm trying to explain - the "semantics" of the elements is completely
> > irrelevant for the purposes of the pairing process, i.e. when we are comparing the "sizes" of the
> > sets.
> >
> > If that person /insists/ that they are still confused due to the different semantics of the
> > elements, what could be said to cheer them up?  Only "just don't be confused! just check out the
> > correspondence - does it work or not?"
> >
> > I'm not sure if this is really what's confusing you, or is it something else?  Since the original
> > quote was discussing the "paradox" of a proper subset of N having "the same size" as N itself,
> > that's what I've focussed on explaining.  I have not really explained about even/odd numbers,
> > because that question is nothing whatsoever to do with the "paradox" being discussed.  (I'd be
> > happy
> > to have a go explaining that too, if you want...)
> >
> >
> > Mike.
> >
>
> I think it you are confused. I and probably the most of the world are interested in the meaning of
> the
> words, not just the appearance of the words, or a specific side of the words you like to see. I am
> not
> the kind of mathematician who believes that mathematics is pure abstract, and nothing do with the
> real world (otherwise, there is no way to verify, and being scientific).
>
>    N      Even
>    0 <--> 0
>    1 <--> 2
>    2 <--> 4
>    3 <--> 6
>    ...
>
> What is the set of even number?
> There are at least two senario of interpretation I think Cantor's set theory
> does not distinguish them apart. The 1st one is when the element of the even
> number is the set N itself, you cannot have a proper 1-1 correspondence with
> this kind of set, so 'cardinality of N' in this case cannot be defined. Try
> drawing a picture to know, I cannot do it here.
>
> The 2nd senario is when Even is 'another set', more likely the right case we
> are really interested in. But such an Even set is also a set of natural number, i.e. N<0,+2>, just a
> different(?) one. So, "the proper subset of N" actually
> refers to this another set of natural number (not the scenario #1 case), which
> is definitely not a proper subset, at least in the sense of a finite set.
>
> I am glad if you, as a mathematician, still want to prech me in the mentality
> that your mathematics is preciser and more usable than mine.
>
>

On Mon, 2024-03-11 at 12:34 +0800, wij wrote:
> On Mon, 2024-03-11 at 12:18 +0800, wij wrote:
> > On Sun, 2024-03-10 at 17:38 +0000, Mike Terry wrote:
> > > On 10/03/2024 01:47, wij wrote:
> > > > On Sat, 2024-03-09 at 17:30 +0000, Mike Terry wrote:
> > > > > On 09/03/2024 11:45, wij wrote:
> > > > > > An example is added about Cantor's set theory in the the section [Paradox Explanation]
> > > > > > https://sourceforge.net/projects/cscall/files/MisFiles/logic_en..txt/download
> > > > > > and thought it may be interested:
> > > > > >
> > > > > > +---------------------+
> > > > > > > Paradox Explanation |
> > > > > > +---------------------+
> > > > > > .....
> > > > > > .....
> > > > > > The number of even number and the number of natural number are equal: Ans:
> > > > > >       ℕ=ℕ<0,+1> and ℕ<0,+2> are isomorphic but the "even numnber" in the two sets
> > > > > >       are semantically different (i.e. the 6 in ℕ<0,+2> is 3 in ℕ<0,+1> ). They
> > > > > >       are two different set of arithmetic systems. Thus, it is confusing to say
> > > > > >       that the number of elements of an infinite set and its proper subset are
> > > > > >       equal.
> > > > > >
> > > > >
> > > > > I agree that the chosen wording above is likely to confuse particulaly non-mathematicians. 
> > > > > That's
> > > > > why when mathematicians talk about infinite sets, they are careful to /define/ the phrases
> > > > > they
> > > > > use
> > > > > to describe them.
> > > > >
> > > > > For example, they typically would not say "The number of even number and the number of
> > > > > natural
> > > > > number are equal", because that would require them to have previously defined "the number
> > > > > of"
> > > > > for an
> > > > > infinite set.  More likely they say one of the following:
> > > > >
> > > > > (a) There is a 1-1 correspondence between the even numbers and the natural numbers
> > > > >       [That is hardly "confusing" to anybody, when the correspondence is demonstrated!]
> > > > >
> > > > > (b) The set of even numbers and the set of natural numbers "have the same cardinality"
> > > > >       [Where "have the same cardinality" is defined as there existing a
> > > > >       1-1 correspondence between the elements of the two sets, i.e. same as (a).]
> > > > >
> > > > > (c) The set of even numbers and the set of natural numbers are "the same size"
> > > > >       [...having /defined/ "the same size" as meaning exactly the same as (a)]
> > > > >
> > > > > This approach avoids ever directly referring to the "number" of elements in the set.
> > > > >
> > > > > Alternatively, perhaps a concept of "cardinal number" has previously been defined, and it's
> > > > > been
> > > > > shown that each set corresponds with a unique cardinal number, such that sets have the same
> > > > > associated cardinal number exactly when (a) above applies.  Then it would also be OK to say:
> > > > >
> > > > > (c) The /cardinality/ of the set of even number equals the /cardinality/ of the
> > > > >       set of natural numbers.
> > > > >
> > > > > Even then I don't think mathematicians would say "The /number/ of even number and the
> > > > > /number/
> > > > > of
> > > > > natural number are equal".  That's just unnecessarily imprecise.
> > > > >
> > > > > Perhaps the only people who would talk about the "number" of elements in an infinite set are
> > > > > non-mathematicians (most of the population!) dabbling in the subject.  Particularly
> > > > > journalists
> > > > > explaining to the general public, and ignorant cranks trying to demonstrate some particular
> > > > > problem
> > > > > with infinite sets (while typically misrepresenting the conventional mathematical
> > > > > standpoint)...
> > > > >
> > > > > Perhaps all this can be classified as a "paradox" about Cantor's set theory, but not in the
> > > > > sense of
> > > > > any problem with the theory.  "Paradox" just in the sense of "unintuitive result when
> > > > > contrasted
> > > > > with finite sets".
> > > > >
> > > > >
> > > > > Regards,
> > > > > Mike.
> > > > >
> > > >
> > > > Stupid is everywhere. Every one can be stupid, every one can be olcott.
> > > > I interpret the response as a perfect demonstration of what the imaginary, stupid
> > > > mathematician would do: Using (lots) more confusing words to cover the fact or ignorance.
> > > >
> > >
> > > If you didn't understand any term I used, just ask about it and I'll explain further.  To be
> > > honest,
> > > I didn't twig that you were the author of the quotation, or that you wanted any explanation for
> > > the
> > > even/odd issue...  (if I had, I'd have replied a bit differently)
> > >
> > > > In the example N<0,+2>, where 6 is actually an odd number. So, what does the
> > > > 'even number' mean? Does it refer to the 'real subset' of the set itself or
> > > > another set? Please provide a clear example that explains what you say in no
> > > > confusing way !
> > >
> > > The quote is about comparing the sizes of sets, right?  So we have two sets:
> > >
> > >    S1 = {1, 2, 3, 4, 5, ...}
> > >    S2 = {2, 4, 6, 8, 10, ...}
> > >
> > > When Cantor/set theory says they are the same size, that is saying that there is a 1-1
> > > correspondence [one-to-one correspondence] between the elements of the set.  Maybe you didn't
> > > understand what that is.  It's a pairing of the elements of the sets, e.g. like this:
> > >
> > >    1  <---->  2
> > >    2  <---->  4
> > >    3  <---->  6
> > >    4  <---->  8
> > >    5  <---->  10
> > >    ...
> > > 1-1 correspondence means every element of S1 appears on the left (above) exactly once, every
> > > element
> > > of S2 appears on the right exactly once, so each element of S1 has a corresponding element in S2
> > > and
> > > vice versa.  I think you would agree that the above does indeed demonstrate such a
> > > correspondence.
> > >
> > > You are asking something about the 3 <----> 6 line, saying that 3 is "an odd number" and 6 is
> > > "even
> > > if considered as the natural number 6, but odd in the sense that it is the 3rd entry in S2 and 3
> > > is
> > > odd".  Or something like that.  So for you, the "semantics" of 3 and 6 as individuals is
> > > different,
> > > so there is some problem with the 1-1 correspondance, which confuses you...
> > >
> > > My response is that we cannot call 3 or 6 even or odd without a lot more "structure" than just
> > > the
> > > bare sets S1 and S2:  as a minimum we need to take into account the addition operations which
> > > are
> > > separate structures from S1 and S2 themselves.  And the key point is that when we are comparing
> > > the
> > > sizes of two sets, WE DISREGARD ALL THAT "EXTRA STRUCTURE" stuff (what I think you refer to as
> > > the
> > > "semantics" of the elements.  We focus just on the individual elements themselves, as though
> > > they
> > > are simply "distinct individuals" in some sense, which are simply to be paired with elements in
> > > the
> > > other set.
> > >
> > > For this pairing process it is of no consequence /what the elements mean/.  Just that they are
> > > correctly paired together.  (Or that such a pairing is not possible.)
> > >
> > > Like when we have two sets
> > >      A = {1, 2, 3}
> > >      B = {A, B, C}
> > > we can match the elements together:
> > >      1  <---->  A
> > >      2  <---->  B
> > >      3  <---->  C
> > > showing that in Cantor world the sets are the same size.
> > >
> > > But then someone comes along and points out "the elements of A are numbers, while the elements
> > > of
> > > B
> > > are letters!  They have different semantics, so it is confusing to say the sets have the same
> > > size!"
> > >
> > > Hopefully you see the point I'm trying to explain - the "semantics" of the elements is
> > > completely
> > > irrelevant for the purposes of the pairing process, i.e. when we are comparing the "sizes" of
> > > the
> > > sets.
> > >
> > > If that person /insists/ that they are still confused due to the different semantics of the
> > > elements, what could be said to cheer them up?  Only "just don't be confused! just check out the
> > > correspondence - does it work or not?"
> > >
> > > I'm not sure if this is really what's confusing you, or is it something else?  Since the
> > > original
> > > quote was discussing the "paradox" of a proper subset of N having "the same size" as N itself,
> > > that's what I've focussed on explaining.  I have not really explained about even/odd numbers,
> > > because that question is nothing whatsoever to do with the "paradox" being discussed.  (I'd be
> > > happy
> > > to have a go explaining that too, if you want...)
> > >
> > >
> > > Mike.
> > >
> >
> > I think it you are confused. I and probably the most of the world are interested in the meaning of
> > the
> > words, not just the appearance of the words, or a specific side of the words you like to see. I am
> > not
> > the kind of mathematician who believes that mathematics is pure abstract, and nothing do with the
> > real world (otherwise, there is no way to verify, and being scientific).
> >
> >    N      Even
> >    0 <--> 0
> >    1 <--> 2
> >    2 <--> 4
> >    3 <--> 6
> >    ...
> >
> > What is the set of even number?
> > There are at least two senario of interpretation I think Cantor's set theory
> > does not distinguish them apart. The 1st one is when the element of the even
> > number is the set N itself, you cannot have a proper 1-1 correspondence with
> > this kind of set, so 'cardinality of N' in this case cannot be defined. Try
> > drawing a picture to know, I cannot do it here.
> >
> > The 2nd senario is when Even is 'another set', more likely the right case we
> > are really interested in. But such an Even set is also a set of natural number, i.e. N<0,+2>, just
> > a
> > different(?) one. So, "the proper subset of N" actually
> > refers to this another set of natural number (not the scenario #1 case), which
> > is definitely not a proper subset, at least in the sense of a finite set.
> >
> > I am glad if you, as a mathematician, still want to prech me in the mentality
> > that your mathematics is preciser and more usable than mine.
> >
> >
>
> So, what is the set of even number as a proper subset and its cardinality?
>
> You seems to believe what Cantor's set theory says and talk about the subset in different context
> (like several others in your math. infinity, limit,dense property,rational number,..., ha)
>
>

On 10.03.2024 18:38, Mike Terry wrote:

>
> The quote is about comparing the sizes of sets, right?  So we have two
> sets:
>
>   S1 = {1, 2, 3, 4, 5, ...}
>   S2 = {2, 4, 6, 8, 10, ...}
>
> When Cantor/set theory says they are the same size, that is saying that
> there is a 1-1 correspondence [one-to-one correspondence] between the
> elements of the set.

And that is a lie as most easily is proved by disproving the
correspondence between natural numbers n/1 and positive fractions:

XOOO...
XOOO...
XOOO...
XOOO...
....

Lossless swaps are lossless.

Regards, WM

On 03/10/2024 10:38 AM, Mike Terry wrote:
> On 10/03/2024 01:47, wij wrote:
>> On Sat, 2024-03-09 at 17:30 +0000, Mike Terry wrote:
>>> On 09/03/2024 11:45, wij wrote:
>>>> An example is added about Cantor's set theory in the the section
>>>> [Paradox Explanation]
>>>> https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download
>>>>
>>>> and thought it may be interested:
>>>>
>>>> +---------------------+
>>>>> Paradox Explanation |
>>>> +---------------------+
>>>> .....
>>>> .....
>>>> The number of even number and the number of natural number are
>>>> equal: Ans:
>>>> ℕ=ℕ<0,+1> and ℕ<0,+2> are isomorphic but the "even numnber" in
>>>> the two sets
>>>> are semantically different (i.e. the 6 in ℕ<0,+2> is 3 in
>>>> ℕ<0,+1> ). They
>>>> are two different set of arithmetic systems. Thus, it is
>>>> confusing to say
>>>> that the number of elements of an infinite set and its proper
>>>> subset are
>>>> equal.
>>>>
>>>
>>> I agree that the chosen wording above is likely to confuse
>>> particulaly non-mathematicians. That's
>>> why when mathematicians talk about infinite sets, they are careful to
>>> /define/ the phrases they use
>>> to describe them.
>>>
>>> For example, they typically would not say "The number of even number
>>> and the number of natural
>>> number are equal", because that would require them to have previously
>>> defined "the number of" for an
>>> infinite set. More likely they say one of the following:
>>>
>>> (a) There is a 1-1 correspondence between the even numbers and the
>>> natural numbers
>>> [That is hardly "confusing" to anybody, when the correspondence
>>> is demonstrated!]
>>>
>>> (b) The set of even numbers and the set of natural numbers "have the
>>> same cardinality"
>>> [Where "have the same cardinality" is defined as there existing a
>>> 1-1 correspondence between the elements of the two sets, i.e.
>>> same as (a).]
>>>
>>> (c) The set of even numbers and the set of natural numbers are "the
>>> same size"
>>> [...having /defined/ "the same size" as meaning exactly the
>>> same as (a)]
>>>
>>> This approach avoids ever directly referring to the "number" of
>>> elements in the set.
>>>
>>> Alternatively, perhaps a concept of "cardinal number" has previously
>>> been defined, and it's been
>>> shown that each set corresponds with a unique cardinal number, such
>>> that sets have the same
>>> associated cardinal number exactly when (a) above applies. Then it
>>> would also be OK to say:
>>>
>>> (c) The /cardinality/ of the set of even number equals the
>>> /cardinality/ of the
>>> set of natural numbers.
>>>
>>> Even then I don't think mathematicians would say "The /number/ of
>>> even number and the /number/ of
>>> natural number are equal". That's just unnecessarily imprecise.
>>>
>>> Perhaps the only people who would talk about the "number" of elements
>>> in an infinite set are
>>> non-mathematicians (most of the population!) dabbling in the
>>> subject. Particularly journalists
>>> explaining to the general public, and ignorant cranks trying to
>>> demonstrate some particular problem
>>> with infinite sets (while typically misrepresenting the conventional
>>> mathematical standpoint)...
>>>
>>> Perhaps all this can be classified as a "paradox" about Cantor's set
>>> theory, but not in the sense of
>>> any problem with the theory. "Paradox" just in the sense of
>>> "unintuitive result when contrasted
>>> with finite sets".
>>>
>>>
>>> Regards,
>>> Mike.
>>>
>>
>> Stupid is everywhere. Every one can be stupid, every one can be olcott.
>> I interpret the response as a perfect demonstration of what the
>> imaginary, stupid
>> mathematician would do: Using (lots) more confusing words to cover the
>> fact or ignorance.
>>
>
> If you didn't understand any term I used, just ask about it and I'll
> explain further. To be honest, I didn't twig that you were the author
> of the quotation, or that you wanted any explanation for the even/odd
> issue... (if I had, I'd have replied a bit differently)
>
>> In the example N<0,+2>, where 6 is actually an odd number. So, what
>> does the
>> 'even number' mean? Does it refer to the 'real subset' of the set
>> itself or
>> another set? Please provide a clear example that explains what you say
>> in no
>> confusing way !
>
> The quote is about comparing the sizes of sets, right? So we have two
> sets:
>
> S1 = {1, 2, 3, 4, 5, ...}
> S2 = {2, 4, 6, 8, 10, ...}
>
> When Cantor/set theory says they are the same size, that is saying that
> there is a 1-1 correspondence [one-to-one correspondence] between the
> elements of the set. Maybe you didn't understand what that is. It's a
> pairing of the elements of the sets, e.g. like this:
>
> 1 <----> 2
> 2 <----> 4
> 3 <----> 6
> 4 <----> 8
> 5 <----> 10
> ...
> 1-1 correspondence means every element of S1 appears on the left (above)
> exactly once, every element of S2 appears on the right exactly once, so
> each element of S1 has a corresponding element in S2 and vice versa. I
> think you would agree that the above does indeed demonstrate such a
> correspondence.
>
> You are asking something about the 3 <----> 6 line, saying that 3 is "an
> odd number" and 6 is "even if considered as the natural number 6, but
> odd in the sense that it is the 3rd entry in S2 and 3 is odd". Or
> something like that. So for you, the "semantics" of 3 and 6 as
> individuals is different, so there is some problem with the 1-1
> correspondance, which confuses you...
>
> My response is that we cannot call 3 or 6 even or odd without a lot more
> "structure" than just the bare sets S1 and S2: as a minimum we need to
> take into account the addition operations which are separate structures
> from S1 and S2 themselves. And the key point is that when we are
> comparing the sizes of two sets, WE DISREGARD ALL THAT "EXTRA STRUCTURE"
> stuff (what I think you refer to as the "semantics" of the elements. We
> focus just on the individual elements themselves, as though they are
> simply "distinct individuals" in some sense, which are simply to be
> paired with elements in the other set.
>
> For this pairing process it is of no consequence /what the elements
> mean/. Just that they are correctly paired together. (Or that such a
> pairing is not possible.)
>
> Like when we have two sets
> A = {1, 2, 3}
> B = {A, B, C}
> we can match the elements together:
> 1 <----> A
> 2 <----> B
> 3 <----> C
> showing that in Cantor world the sets are the same size.
>
> But then someone comes along and points out "the elements of A are
> numbers, while the elements of B are letters! They have different
> semantics, so it is confusing to say the sets have the same size!"
>
> Hopefully you see the point I'm trying to explain - the "semantics" of
> the elements is completely irrelevant for the purposes of the pairing
> process, i.e. when we are comparing the "sizes" of the sets.
>
> If that person /insists/ that they are still confused due to the
> different semantics of the elements, what could be said to cheer them
> up? Only "just don't be confused! just check out the correspondence -
> does it work or not?"
>
> I'm not sure if this is really what's confusing you, or is it something
> else? Since the original quote was discussing the "paradox" of a proper
> subset of N having "the same size" as N itself, that's what I've
> focussed on explaining. I have not really explained about even/odd
> numbers, because that question is nothing whatsoever to do with the
> "paradox" being discussed. (I'd be happy to have a go explaining that
> too, if you want...)
>
>
> Mike.
>

On 03/12/2024 07:52 PM, Ross Finlayson wrote:
> On 03/10/2024 10:38 AM, Mike Terry wrote:
>> On 10/03/2024 01:47, wij wrote:
>>> On Sat, 2024-03-09 at 17:30 +0000, Mike Terry wrote:
>>>> On 09/03/2024 11:45, wij wrote:
>>>>> An example is added about Cantor's set theory in the the section
>>>>> [Paradox Explanation]
>>>>> https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download
>>>>>
>>>>>
>>>>> and thought it may be interested:
>>>>>
>>>>> +---------------------+
>>>>>> Paradox Explanation |
>>>>> +---------------------+
>>>>> .....
>>>>> .....
>>>>> The number of even number and the number of natural number are
>>>>> equal: Ans:
>>>>> ℕ=ℕ<0,+1> and ℕ<0,+2> are isomorphic but the "even numnber" in
>>>>> the two sets
>>>>> are semantically different (i.e. the 6 in ℕ<0,+2> is 3 in
>>>>> ℕ<0,+1> ). They
>>>>> are two different set of arithmetic systems. Thus, it is
>>>>> confusing to say
>>>>> that the number of elements of an infinite set and its proper
>>>>> subset are
>>>>> equal.
>>>>>
>>>>
>>>> I agree that the chosen wording above is likely to confuse
>>>> particulaly non-mathematicians. That's
>>>> why when mathematicians talk about infinite sets, they are careful to
>>>> /define/ the phrases they use
>>>> to describe them.
>>>>
>>>> For example, they typically would not say "The number of even number
>>>> and the number of natural
>>>> number are equal", because that would require them to have previously
>>>> defined "the number of" for an
>>>> infinite set. More likely they say one of the following:
>>>>
>>>> (a) There is a 1-1 correspondence between the even numbers and the
>>>> natural numbers
>>>> [That is hardly "confusing" to anybody, when the correspondence
>>>> is demonstrated!]
>>>>
>>>> (b) The set of even numbers and the set of natural numbers "have the
>>>> same cardinality"
>>>> [Where "have the same cardinality" is defined as there existing a
>>>> 1-1 correspondence between the elements of the two sets, i.e.
>>>> same as (a).]
>>>>
>>>> (c) The set of even numbers and the set of natural numbers are "the
>>>> same size"
>>>> [...having /defined/ "the same size" as meaning exactly the
>>>> same as (a)]
>>>>
>>>> This approach avoids ever directly referring to the "number" of
>>>> elements in the set.
>>>>
>>>> Alternatively, perhaps a concept of "cardinal number" has previously
>>>> been defined, and it's been
>>>> shown that each set corresponds with a unique cardinal number, such
>>>> that sets have the same
>>>> associated cardinal number exactly when (a) above applies. Then it
>>>> would also be OK to say:
>>>>
>>>> (c) The /cardinality/ of the set of even number equals the
>>>> /cardinality/ of the
>>>> set of natural numbers.
>>>>
>>>> Even then I don't think mathematicians would say "The /number/ of
>>>> even number and the /number/ of
>>>> natural number are equal". That's just unnecessarily imprecise.
>>>>
>>>> Perhaps the only people who would talk about the "number" of elements
>>>> in an infinite set are
>>>> non-mathematicians (most of the population!) dabbling in the
>>>> subject. Particularly journalists
>>>> explaining to the general public, and ignorant cranks trying to
>>>> demonstrate some particular problem
>>>> with infinite sets (while typically misrepresenting the conventional
>>>> mathematical standpoint)...
>>>>
>>>> Perhaps all this can be classified as a "paradox" about Cantor's set
>>>> theory, but not in the sense of
>>>> any problem with the theory. "Paradox" just in the sense of
>>>> "unintuitive result when contrasted
>>>> with finite sets".
>>>>
>>>>
>>>> Regards,
>>>> Mike.
>>>>
>>>
>>> Stupid is everywhere. Every one can be stupid, every one can be olcott.
>>> I interpret the response as a perfect demonstration of what the
>>> imaginary, stupid
>>> mathematician would do: Using (lots) more confusing words to cover the
>>> fact or ignorance.
>>>
>>
>> If you didn't understand any term I used, just ask about it and I'll
>> explain further. To be honest, I didn't twig that you were the author
>> of the quotation, or that you wanted any explanation for the even/odd
>> issue... (if I had, I'd have replied a bit differently)
>>
>>> In the example N<0,+2>, where 6 is actually an odd number. So, what
>>> does the
>>> 'even number' mean? Does it refer to the 'real subset' of the set
>>> itself or
>>> another set? Please provide a clear example that explains what you say
>>> in no
>>> confusing way !
>>
>> The quote is about comparing the sizes of sets, right? So we have two
>> sets:
>>
>> S1 = {1, 2, 3, 4, 5, ...}
>> S2 = {2, 4, 6, 8, 10, ...}
>>
>> When Cantor/set theory says they are the same size, that is saying that
>> there is a 1-1 correspondence [one-to-one correspondence] between the
>> elements of the set. Maybe you didn't understand what that is. It's a
>> pairing of the elements of the sets, e.g. like this:
>>
>> 1 <----> 2
>> 2 <----> 4
>> 3 <----> 6
>> 4 <----> 8
>> 5 <----> 10
>> ...
>> 1-1 correspondence means every element of S1 appears on the left (above)
>> exactly once, every element of S2 appears on the right exactly once, so
>> each element of S1 has a corresponding element in S2 and vice versa. I
>> think you would agree that the above does indeed demonstrate such a
>> correspondence.
>>
>> You are asking something about the 3 <----> 6 line, saying that 3 is "an
>> odd number" and 6 is "even if considered as the natural number 6, but
>> odd in the sense that it is the 3rd entry in S2 and 3 is odd". Or
>> something like that. So for you, the "semantics" of 3 and 6 as
>> individuals is different, so there is some problem with the 1-1
>> correspondance, which confuses you...
>>
>> My response is that we cannot call 3 or 6 even or odd without a lot more
>> "structure" than just the bare sets S1 and S2: as a minimum we need to
>> take into account the addition operations which are separate structures
>> from S1 and S2 themselves. And the key point is that when we are
>> comparing the sizes of two sets, WE DISREGARD ALL THAT "EXTRA STRUCTURE"
>> stuff (what I think you refer to as the "semantics" of the elements. We
>> focus just on the individual elements themselves, as though they are
>> simply "distinct individuals" in some sense, which are simply to be
>> paired with elements in the other set.
>>
>> For this pairing process it is of no consequence /what the elements
>> mean/. Just that they are correctly paired together. (Or that such a
>> pairing is not possible.)
>>
>> Like when we have two sets
>> A = {1, 2, 3}
>> B = {A, B, C}
>> we can match the elements together:
>> 1 <----> A
>> 2 <----> B
>> 3 <----> C
>> showing that in Cantor world the sets are the same size.
>>
>> But then someone comes along and points out "the elements of A are
>> numbers, while the elements of B are letters! They have different
>> semantics, so it is confusing to say the sets have the same size!"
>>
>> Hopefully you see the point I'm trying to explain - the "semantics" of
>> the elements is completely irrelevant for the purposes of the pairing
>> process, i.e. when we are comparing the "sizes" of the sets.
>>
>> If that person /insists/ that they are still confused due to the
>> different semantics of the elements, what could be said to cheer them
>> up? Only "just don't be confused! just check out the correspondence -
>> does it work or not?"
>>
>> I'm not sure if this is really what's confusing you, or is it something
>> else? Since the original quote was discussing the "paradox" of a proper
>> subset of N having "the same size" as N itself, that's what I've
>> focussed on explaining. I have not really explained about even/odd
>> numbers, because that question is nothing whatsoever to do with the
>> "paradox" being discussed. (I'd be happy to have a go explaining that
>> too, if you want...)
>>
>>
>> Mike.
>>
>
>
> I think what confuses people about the disconnect between
> cardinality as "size" and density as "size", in terms of,
> well, in terms of "absolute size" or "relative size",
> is that their modular notion of density as relative size,
> for numbers, is more apropos under most all conditions
> that aren't about quantifying over the entire range of
> the numbers, only, you know, boundedly.
>
>
> Like hopscotch, when skipping squares, half the squares
> are skipped, there are twice as many as are skipped.
>
> Saying "cardinality is absolute size and their size
> is only the same in unbounded hopscotch", has skipped
> all the finite cases.
>
>
> Then it's similar with OUTPACING for sets generally,
> transitively proper subsets get smaller, it's a "size"
> relation, cardinality is only a specific size relation,
> that happens to be apropos for Aleph_0 and c (2^Aleph_0).
>
> In this same thread though split between sci.logic and
> comp.theory, it's sort of described that while set theory
> is a particular formal theory of great interest, that
> number theory's objects are numbers and set theory's
> objects are sets, and modeling each in terms of the
> other, leads to "talking past each other", was the phrase.
>
>
> Other aspects of "naive" set theory, are that "naive"
> set theory is only set-builder notation, and that is all,
> then it includes any quantification at all.
>
> So, in naive set theory, one might write "the set of sets",
> it's not a set in ZF. The naive set theory is rather
> ignorant any restrictions of comprehension, where ZF's
> restrictions of comprehension are that:
> P e Q, or, Q e P, or, neither, and: not both", then
> the combination of expansion and restriction:
> "there's an infinite set, and it's ordinary", and,
> "there an empty set, and it's unique", those are
> its rules, otherwise naively that expansion of comprehension
> can never go wrong, and for most all sorts usual very
> simple definitions up after set-builder, is that set-builder
> is all that students need that then their results like
> "in the integers Z" or "in the real numbers R" and so on
> have that half the Z's count or R's length of an interval of
> those is in either half of a partition in the middle.
>
> That then founds "density" the most usual notion of,
> "size of subsets, or relative size, with respect to numbers",
> where cardinality then is its own "absolute" notion of
> size as it were, that only makes sense when talking
> about whether two sets have the same cardinal or not.
>
>
>
> Of course, this isn't even beginning to address the
> idea of the continuum limit of naturals n/d as d->oo,
> line-continuity,
> or about what goes into classifying all the functions
> like Fourier analysis that result their intersection
> in the continuum limit some kind of signal-continuity,
> vis-a-vis the much, much later formal treatment for
> field-continuity, or the standard way that provides
> the standard model for R, than the naive, which is
> built on the properties of set-like operations,
> and the laws of arithmetic.
>
>
> Trans-finite numbers are pretty much irrelevant
> to most students.
>
> That doesn't at all change what they are, cardinal
> numbers, it's just that, one need not necessarily
> the attachment to them as necessarily fundamental
> that all thought processes go through them, when
> instead, it's "only" the realm of transfinite induction.
>
> Naive set theory and set-builder notation of course
> is great and natural and clear geometrically.
>
>