After looking through the memories of the fallen cardinal-level mysterious palms of the three worlds, Mu Cang learned about it.
The endless mysterious palms entrenched in this vast territory community are actually not a group of dragons without a leader fighting on their own, but there is a supreme leader in a substantial sense.
This person, who is in charge of all military affairs and power of the territorial community in this area, is called the "Governor of the Frontier" in the position system of the leader's civilization.
This title, as the name suggests, refers to the governor who guards the border areas.
And this governor, in the memories of those three mysterious palms, is exactly a genuine unreachable cardinal level master.
At the same time, this governor is also the only unreachable cardinal-level mysterious palm in this territorial community.
Moreover, it is a strongly unreachable cardinality.
Except for this governor, all other Mysterious Palms are at the world base level, or in other words, they are all in the huge base category of the world base.
This is completely understandable.
Regardless of the majesty and distance of the unreachable cardinal number, you must know that within the category of the world cardinal number alone, it is completely possible to divide endless levels, and the gaps between each level are also infinite and boundless.
So how big is this so-called "infinite, countless and boundless"?
It can be understood this way.
If we say, starting from the smallest infinity——??, the journey to the first world base number WC is so far and long.
So the journey from the first world base WC to the 1-world base WC is equally long and far away, or even farther and longer.
Why is this so?
Because the essence of 1-world cardinality WC is that on the basis of the first world cardinality WC axiom model introduced into a certain ZFC axiom system model, through various extremely complex methods, it can be re-encapsulated into a ZFC model.
high.
In the same way, the path from the n-world base to the n 1-world base, or the path from the x-world base to the x 1-world base, or even the path from the k-world base to the k 1-world base
The journey is equally far and long.
These explanations and analogies are indeed somewhat difficult to understand at first glance.
So to put it more thoroughly, any large cardinality axiom actually far exceeds the proving ability or jurisdiction of the ZFC axiom system model itself.
If it is described in a fairy-tale style, any big cardinal number is an innate chaotic demon god that is too powerful, so powerful that if it only relies on the ZFC axiom system's own capabilities, there is no way it can be conceived.
Therefore, only after being settled by the Chaos Demon named Big Cardinal, can the Von Neumann Universe V in the 'blank slate' state be able to achieve higher strength and possess more colorful properties.
In fact, for the countless finite, infinite, and super-finite life forms, Cantor's absolute infinity is approximately equal to the so-called "omniscience and omnipotence" in their cognitive scope.
However, its consistency strength is equal to or even superior to Cantor's absolutely infinite ZFC model. After having any large cardinality axiom, its strength can actually skyrocket to a higher level that is indescribable.
From this, we can see how terrifying the strength of that large base number is, so terrifying that even words such as Cantor's absolute infinite multiples, which far exceed the so-called "omniscient and omnipotent" level, are simply not enough to describe it.
In short, when the world cardinality WC introduces the W function and then uses the substitution axiom of ZFC, and then continuously improves the level by performing sup operations similar to the ? function, the universal mathematical universe that contains and accommodates the world cardinality WC will also
They also keep climbing up the ranks together.
When this kind of advancement is truly presented in the concrete physical world, the digital realm will be like a tower of Babel. While continuously expanding the foundation, it will also continue to crazily pile up the floors, and expand and pile up.
The level of difficulty is always so terrifying.
But this climbing method also has its limit. This limit is the fixed point of the world base, which can also be called the "world point" of the W function.
On top of this, there are also world cardinal fixed points that are far from being able to describe their specific number even with the word "infinite".
But these world cardinality fixed points will be intercepted below by k=k?world cardinality...that is, the "great world cardinality".
No matter how huge the gap between all the previous world bases is, they are all equally small to the great world bases.
Because the common mantissas of these world cardinal numbers are all only w.
As for the so-called "cotail", it is an important mathematical concept in set theory. It is mainly used to describe the characteristics and sophistication of unbounded subsets of well-ordered sets and sequences.
To put it bluntly, when many increasing sequences can only use ordinal numbers below a, how many items are needed to reach them, so the term "gradient" can also be used to refer to them.
And if we expand the concept that all world cardinals below the great world cardinal have a final degree of w, that is, for all n∈N, the sequence of the minimum ∑n correct cardinal number is the next basic sequence of k with length w,
At the same time, for any n, ∑n 1 can be used to describe the correct base of a certain ∑n, so its strength is within the scope of the ZFC axiomatic model.
However, for the basic column as a whole, there is no certain ∑m statement that can describe all ∑n. Because there is no natural number larger than all natural numbers, the basic column of k cannot be defined within Vk, so it cannot be used as a set.
Due to the substitution axiom, this basic column must be defined outside the ZFC model, that is, in Vk 1.
In short, what is above a series of world cardinal fixed points is the great world cardinal number. However, there are also endless and infinite fixed points of the W function on the great world cardinal number, and these fixed points are extremely far away from each other.
points also have the same co-mantissa.
Therefore, after reaching this level, the co-mantissa can also be regarded as a measure of strength between different levels in a very rough way.
And the 'closest' higher co-mantissa level to the series of all world cardinals of this co-final w is w1 which is equipotential with ??.
On top of this, there are w? that is of equal potential to ??, w?? of equal potential to ???, w???
It is so huge that it has no edges at all.
These various world cardinal numbers with different co-mantissas are usually named with various complex prefixes or suffixes.
Moreover, within the huge 'territory' that is 'ruled' by each of these co-mantissas of all levels, there will also be endless, endless, endless, and indescribably terrifying world bases among each other.
describe the huge gap.
And if you want to cross these vast distances, the mathematical concept of the so-called "unbounded closed set" will be involved.
Regarding this concept, there is also a simpler prepositioned concept called "unbounded set".
To illustrate this concept, for example, the natural numbers in the category w are unbounded in w, and since w = N, N is an unbounded and non-proper subset of w. (For the specific definition of the concept of "unbounded", see Chapter 677.
)
Since there is ‘non-truth’, then there must be ‘truth’.
For example, for any n∈w, there is still n 1∈w, and there is no largest natural number, so all positive even numbers are the true unbounded subset of w.
This concept is relatively simple, but the concept of "unbounded closed set" above this requires more consideration... No, it is much more complicated.
Or give an example.
For example, if c is an unbounded subset of x, for all limit ordinal numbers it is a
