If you do accept the ZF axioms, and the entire philosophy of axiomatizations, then you have to find out the exact error in the fully rigorous proof as can be found for example on metamath:
canth - Metamath Proof Explorer
Thanks very much. Your post is very nice and comforting. Your site is great, but I do not understand the mathematical language it uses. So, I use natural language that Cantor surely used in his time.
Below is a short summary of the first error I have found in Cantorâ€™s theorem for discussion here.
Cantor uses a proof by contradiction, which is the following.
The proposition to be proved P: A list cannot contain all subsets of â„•.
1. Assume that P is false. That is: a list L contains all subsets of â„•.
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.
See section â€œRecall of the proofâ€ of Â«
Analysis of the proof of Cantor's theorem Â»
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 â€œCase of infinite setâ€ of Â«
Analysis of the proof of Cantor's theorem Â» 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 Lâ€™ contains the diagonal.
The length of the diagonal equals the length of the list Lâ€™ and the length of â„• which is the width of the list Lâ€™. So, the length of L equals the length of â„•. In this case, the proposition â€œA list cannot contain all subsets of â„•â€ is not correct, because in the derivation the use of the property â€œselfish or non selfishâ€ imposes the length of the list L to equal the length of â„•. 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 â„• cannot be put in the list L with the length of â„•, subsets of â„• are uncountable. But is â„š countable? â„š is created by one â„• in x-axis and another in y-axis. In this case, â„š is a list whose length is a free variable not limited by â„• of its axis. Must the length of â„š equal the length of its axis?
PK