Re: A paradox about Cantor's set theory

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