I think it’s better to publish the Infinity series in two parts. The first part is almost finished. There is only one more section to add. There are new things in this section that haven’t been discussed yet. So I post them here:
Some issues about the countable set and its proof were mentioned previously. For example, the infinity mechanism cannot be supposed to be given. Cantor implicitly supposed that, in the 1-1 mapping between N and R, all natural numbers have been used and it’s not possible to construct a new natural number by adding 1 to the last element. But, the issue is that, at the same time he supposed that all real numbers have not been used. It may seem that he proved that all R have not been used by constructing a new real number. This is not true. To clarify, assume this is the list of 1-1 mapping between R and N:
1 → R_1
2 → R_2
…
N → R_N
By assumption, N is a finite number, in fact, the very last one. But, again by assumption, N+1 diverges because N is supposed to be the last finite number. So the assumption contradicts a basic property of numbers. It is assumed N is a finite number, yet is a special type of finite number such that N+1=inf. Assumptions shouldn’t contradict basic properties. The infinity axiom assumes that Cantor’s assumption is valid, that is, the axiom says that there is a set that contains all natural numbers plus infinity. But axioms shouldn’t contradict other properties. For example, In Euclid’s framework, exactly one line passes through two points. This cannot be proven but this doesn’t contradict any other assumption or a property derived from other axioms. The infinity axiom contradicts the fact that N+1 is finite if N and 1 are both finite. If N is the very last natural number, it should also possess another property not found in any other natural number so one can’t be added to N. The infinity axiom as a result is more than one assumption. It should also assume there is one or more natural numbers that are different from other natural numbers but still they are in the list of natural numbers. More accurately, the infinity axiom assumes there is at least one mechanism that pushes natural numbers to the inf zone without providing insight into the mechanism or mechanisms deployed. The cantor proof assumes a 1-1 mapping between two different infinity mechanisms, one for natural numbers and the other for R, can be established. It’s unclear if the notion of 1-1 mapping makes sense here at all. This exposes the issue with the proof. This is a proof by contradiction, yet the negation of the initial assumption is undefined. In fact, it’s unclear whether the assumption itself is a valid expression.
UPDATE 1:
Proof by contradiction is tricky. Suppose 1-1 mapping between two mechanisms can be established. Cantor’s proof shows that the initial assumption is incorrect. Imagine that in reality there are more N than R. So the initial assumption of the proof is incorrect. The diagonal method shows that the assumption is incorrect, but it doesn’t say there are more R than N. It’s true that a new R has been constructed. but the ability to construct a new R implies that the initial assumption was wrong. It could very well mean that in reality there were more N than R.
UPDATE 2:
When an assumption is wrong, it could be contradicted in all sorts of nonsensical ways. It doesn’t really matter how it contradicts, the contradiction itself matters.