Re: A paradox about Cantor's set theory

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 !