Below is a short summary of the first error I
have found in Cantor’s theorem, given in reply
in mymathforum.com for discussion.
Cantor uses a proof by contradiction, which is
the following.
The proposition to be proved P: A list cannot contain all subsets
of ℕ.
1. Assume that P is false. That is: a list L contains all
subsets of ℕ.
2. A subset K is created. It is shown that K is not in L,
which shows P is true.
3. P cannot be true and false at the same time, So, P is true.
The scheme of logic is:
1. P is assumed false
2. P is shown true.
3. Then P must be true.
This logic holds if P is correctly stated.
What if the statement of P is incorrect? In this case, will the proof fall
apart? The error I have found is that the used list in P is limited in length
by the creation of K.
In section “Case of infinite set” of « Analysis of the proof of Cantor's theorem » I have shown
that for a subset to be selfish or non selfish, the binary string of the subset
must have the diagonal bit which equals 1 or 0. As the list L contains only
subsets that are selfish or non selfish, the binary list L’ contains the
diagonal.
The length of the diagonal equals the length of the list L’ and
the length of ℕ which is the width of the list L’. So, the length of L equals the length of ℕ. In this case, the proposition “A list cannot contain all
subsets of ℕ” is not correct, because
in the derivation the use of the property “selfish or non selfish” imposes the length of the
list L to equal the length of ℕ. So, the length of the list
in the proposition P is not free of constraint, but is imposed by another
element in the same context.
One can argue that because all subsets of ℕ cannot be put in the list L with the length of ℕ, subsets of ℕ are uncountable. But is ℚ
countable? ℚ is created by one ℕ in
x-axis and another in y-axis. In this case, ℚ is a list whose length is
a free variable not limited by ℕ of its axis. Must the length of ℚ equal the length of its
axis?
