Cantor's Logic (10) The Fundamental Series contained in a Transfinite Ordered Aggregate

I found the following points worth further analysis in sections 10 and 11.

The Imploded Number Line
"An ascending fundamental series {a_v} and a descending one {b_v} are said to be "coherent," in signs
(5){a_v} || {b_v},
if (a) for all values of v and u, 
a_v -< b_u,
and (b) in M exists at most one (thus either only one or none at all) element m₀ such that, for all v's,
a_v -< m₀ -< b_v." (p.130)

I find this a very strange affirmation. Take the infinite series containing negative and positive integers. Any point can be considered as the origin, and will therefore follow all those less than itself, and precede all those higher than itself. So, what does Cantor mean, if not that?
A less obvious use of the principle would mean that there can be no two origins that are distinct and equal at the same time.
We can choose any origin we want, but no other origin will give the same result. Not in the same series at least.

This is the birth of the "imploded universe" or "multi-dimensional objects" I mentioned earlier (9 and 15.4). Note that Cantor plays on the one dimensionality of the fundamental series on one hand, and the multi-dimensional character of the set M containing all those series on the other hand. That allows him to switch his perspective without explicitly declaring it, like I showed earlier. [See also (*) below.]

"B. Two fundamental series proceeding in the same direction of which one is part of the other are coherent.
If there exists in M an element m which has such a position with respect to the ascending fundamental series {a_v} that :
(a) for every v
a_v -< m_O ,
(b) for every element m of M that precedes m₀ there exists a certain number v₀ such that
a_v >- m, for v >= v₀" (p.130-131)

It should look like this:

__m___v₀___a_v___m₀______

"then we will call m₀ a "limiting element (Grenzelement) of {a_v} in M " and also a "principal element (Hauptelemenf) of M." In the same way we call m₀ a "principal element of M" and also "limiting element of {b_v} in M" if these conditions are satisfied :
(a) for every v
b_v >- m₀,
(b) for every element m of M that follows m₀ exists a certain number v₀ such that
b_v >- m, for every v >= v₀." (p.131)

__m___v₀___a_v___m₀___v₀___b_v

The same variable v is used for elements preceding and elements following m₀, which is a little bit confusing.
This concerns obviously the one-dimensional series, what Cantor calls the "fundamental series".

The following statement is in fact what the whole is all about:

(*) "A fundamental series can never have more than one limiting element in M ; but M has, in general, many principal elements."

What remains astounding is how such a set, which reminds me of the magical bucket our djinn used, can even be possible. The assumption of actual infinity alone makes it conceivable: all possible infinite series together in one bucket, or set.
Unlike the djinn with his bucket out of which the magic creature could only pull one single digit at a time, Cantor is able to pull any order he wants out of the set, or point at its inherent anarchy when needed.
Cantor never speaks (starting at section 9 ) of type eta other than a one dimensional object like the number line. The property of "everywhere dense" is also understood as local black holes, as it were. Wherever you stop you risk drowning in the infinite depths of each point. This way, the idea that any element has an infinity of elements preceding it keeps its one-dimensionality.
But it is a fake one-dimensionality. You enter each time an infinite universe, which makes the number line as complex as an infinite cluster of galaxies. 
Chaos is luckily the savior of rationality. Because there are apparently no rules of precedence, the mind is free to impose its own order. Otherwise it would go from one preceding element to the next, never sure if it is still on the same series, or whether it has inadvertently switched dimensions. The mind would for ever be trapped in the same diagonal maze Cantor had devised. A rational trap, that's for sure, and for that very reason, much more dangerous than chaos which can be sculpted at will.
Cantor never explains to us how it is that we are able to solve this modern version of Zeno's paradox. How come we go so easily from one number to the other while counting for instance? Apparently, the mathematician's freedom is in itself enough to keep him from drowning in all those little black holes. You have to choose to enter the diagonal maze, you cannot just tumble into it, like Alice does in the rabbit hole.
It is also important to realize that the rabbit hole does not explain the breakdown of logic. Infinity does not necessarily mean that n+n=n, or n.n=n. There is no inherent logical necessity between both ideas. Nothing logical prevents us from going down the infinite series of irrational numbers one number at a time, and in any direction. Each step would be in itself a logical step, easily explainable by its history. That our mind would eventually implode is a human idiosyncrasy, not the end of logic.
And that is exactly what Cantor wants from us, to give up our belief that 1+1=2 is valid even in the rabbit hole.
Actually, Cantor only wants us to suspend that belief, not give it up entirely. Otherwise we could never get from one level of infinity to the other. "1+1=2" needs sometimes to be true. The whole point is to convince us that there is a logical pattern to the on and off switching of our very logical intuitions.
I am afraid, I am still unconvinced.