Re: A paradox about Cantor's set theory

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.