Attributes are infinitely increasing, I dominate multiple

When the infinite mathematicians of the infinite universe, from ancient times to the present, derived and proved the ultimate mathematical concept, they faced another urgent problem.

That is, how to give this ultimate concept, which seems to be the core foundation of the entire mathematical system, an appropriate name?

Strangely enough, after this idea came up, the infinite mathematicians of infinite time and space inexplicably and coincidentally flashed a name in their hearts... [Mu Cang] out of thin air.

The appearance of this name is very natural, almost like breathing.

So the countless mathematicians also very naturally used the language and writing of their own civilization to name the ultimate concept that supports the various axioms of the entire mathematical system... "Mu Cang Axiom".

And now, the true ontology or origin of this "ultimate axiom".

Mu Cang, the almighty God of the world cardinality level standing at the pinnacle of the mathematical axiom system of the Gödel-constructable universe, was reading the huge memory information of the three Xuanzhangs.

In fact, Mu Cang had never experienced the so-called configuration imitation and Xuansui transformation, let alone any ontological reconstruction and transformation and sublimation.

Under the mysterious power of the heaven-defying skill [Wu Jue Mi Ce], He directly skipped all the processes and procedures and was promoted to a large cardinality life form - a Xuansui-level master in one step.

Although the Xuansui core that Mu Cang now possesses is only the smallest large cardinality - the world cardinality axiom logic configuration.

But the large cardinality is the large cardinality.

Even if it is only the world cardinality, it is not comparable to any existence below the large cardinality level, and there is no comparability at all.

In fact, the imagination of intellectual life is extremely poor.

It is so poor that even the smallest cardinality, such as the world cardinality, cannot be described in any concrete way with any figurative words. It can only be described vaguely through a side summary mixed with mathematical language.

So, how to describe or describe the hugeness of the world cardinality?

This needs to be explained from the beginning.

From the perspective of set theory, infinity is a limit, not a specific value.

Therefore, the comparison between infinities of different series is always the size of the so-called "potential".

For example, the number of all natural numbers is defined as the 0th level infinity n through the infinity axiom. On this basis, the number of all subsets of the 0th level infinity can be defined as the higher level 1st level infinity 2n through the power set axiom.

So a series of infinities of various levels can be constructed by analogy, such as 2nd level infinity, 10th level infinity, 100th level infinity, etc. ... until infinite level infinity.

The size of these various infinities of various series compared with each other is also the size of "potential".

Then we can also let the infinite level infinity be k, and then construct a higher level k-level infinity, k-level infinity-level infinity, k-level infinity-level...k infinity-level infinity, and so on.

In this way, we go through countless infinite cycles over and over again, and through the? function and △? formula, we can get the? fixed point.

Above it, there are still more and larger? fixed points, as well as endless pa fixed points, and infinite PA fixed points.

Then, at the top of the various fixed points that are so numerous that they cannot be described by infinite, infinite, countless and endless, is the ∑2 world cardinality that cannot be described by the so-called supreme heaven, the supreme kingdom of God, the final other shore... all kinds of adjectives.

Note that it is the ∑2 world cardinality, not the world cardinality. The two are completely different concepts.

For the ∑1 world cardinality, if α is a power-admissible cardinality, then Vα is a model of ZFC.

〖ZFC〗 means that the substitution axiom of ZFC is completely restricted to the range of the ∑1 formula.

As for the ∑1 formula, it is a 1st-order existential proposition with only one unbounded existential quantifier at the beginning.

The so-called "unbounded" means that it will be larger than any given bounded value, and in order to reach the ∑1 world cardinality, all recursive operations of the Aleph function need to be closed.

As for the ∑2 world cardinality above it, it is much more complex and huge, because the beginning of its mathematical formula is an unbounded existential proposition linked to an unbounded universal proposition.

From the perspective of set theory, if α is a ∑2 world cardinality, then as long as α has some local properties, there must be an unbounded multiplicity kltα that also has the same local properties as α.

At the same time, the so-called local properties of α mean that this property can be witnessed only in the segment before Vα, which does not and does not need to involve higher-level fields.

If it involves higher fields, it is a global property.

Another point is that perhaps many intellectual beings have learned about the so-called "absolute infinity" mentioned by Cantor from books or other people.

In fact, if we follow the idea of ​​naive set theory, which is a relatively rigid and a priori "domain", the basic description of the cardinality of the ∑2 world can fully meet all the sufficient conditions required by Cantor's absolute infinity.

Note that the absolute infinity mentioned here is not the kind of broad and vague absolute infinity that tends to be theological, philosophical, or personal, but the absolute infinity of naive set theory.

Naive set theory...or the essence of Cantor's absolute infinity is that any property can be shared by all other smaller infinite bases.

To break it down in more detail, the simple set theory defined by Cantor is absolutely infinite, which means that there is a basis --\u003e\u003e

Count Ω.

And no matter which type of unformulated properties are used to describe the abstract concept of "big", Ω is not the first infinite base to have this property, but there are already Ω smaller infinite bases than Ω. also possesses such properties.

Therefore, Ω cannot be constructed from the bottom up using any infinite base number smaller than Ω, so Ω is absolutely infinite and is an indescribable and unsurpassable God.

This kind of "absolute infinity Ω" with a somewhat theological nature naturally does not conform to the thinking of many intellectual beings, the kind of absolute infinity that is almost equated with the so-called omniscience and omnipotence.

But it can also be seen from this that the Σ2 world base is profound and huge.

In short, the Σ2 world cardinality can be regarded as "simple absolute infinity".

And since there is 2, then naturally there will be 3, 4, and 5.

Therefore, on top of the Σ2 world cardinal number, there are also Σ36 world cardinal numbers, Σ9999 world cardinal numbers, and Σ12 world cardinal numbers that can take all natural numbers.

Each of this series of Σ world cardinal numbers is far...far beyond the endless...endless and infinite levels of the Σ2 world cardinal numbers that are equivalent to simple absolute infinity.

Strictly speaking, all these Σ world cardinal numbers are the supremum of ω Σ n world cardinal numbers, and are the real ‘minimum’ large cardinal numbers – the world cardinal numbers.

This is the level of life Mu Cang is currently at.

At this level, from the perspective of simple set theory, He has far surpassed Cantor's absolute infinite infinite number.

The magnitude of its transcendence is so huge that it is simply indescribable to use Cantor's absolute infinity, Cantor's absolute infinity, and Cantor's absolute infinity.

As for the higher and more distant unreachable base, it is completely beyond the sky, and any world base cannot be compared with it.

"so……"

Mu Cang, who was sitting on the top of Gödel's constructible universe, whispered with deep eyes, "Is this vast territory of the universe just a corner of an inevitable country?"

In the memories of the three fallen Xuanzhangs, Mu Cang gained insight into an astonishing fact.

This is a territorial community composed of an infinite number of various digital realms that are loosely linked to each other. It is surprisingly just a corner on the edge of the sphere of influence of the Taoist clan.

Therefore, there are also an infinite number of mysterious palms, which are scattered at the top of the axiom systems of the numerical and logical territorial backgrounds in this community, and are in charge of all things in each territory.

The reason why so many mysterious palms exist here is very consistent with Mu Cang's previous guess.

It is this territorial community that is indeed a front line of battle between the inevitable state and the free state.

To be more precise, this is a low-intensity war zone between the two countries.

As for why the two major groups, the Taoist and the Undecided, are so incompatible, the reason is closely related to the basic concepts upheld by the two groups, as well as the development and survival of the two groups.

For example, the Taoist family, their consistent idea or dream is to give an absolutely certain truth value structure to all mathematical formal systems with meaningful or meaningless logical structures, and thereby give all mathematical formal systems There is an ultimate good order of all things with sufficient strength of proof theory, and then the absolute perfect unity of all concrete things and all abstract things, as well as all things under the two, above the two, and on the side of the two.

The Undecideds are the complete opposite. They hate all so-called meanings and truth values, and therefore reject all forms and structures.

Because those who are undecided believe that truth represents bondage, meaning also represents bondage, and structure and form are the chains and torture instruments that realize bondage.

In their view, all constraints and all the concreteness and abstraction derived from them are the opposite of freedom.

If you want to gain ultimate freedom, you must smash all structures and forms, and then disintegrate all truth values ​​and all meanings, so that everything will fall into endless uncertainty and meaninglessness.

Only in this way, only by allowing everything to turn into true chaos, can we have true freedom.

And that free country controlled by the undecided family did exactly this.

Mu Cang saw in the memories of these three mysterious palms that the undecided group of people suddenly appeared in the Lost Abyss at a certain moment. Since their appearance, they have been working tirelessly to conquer one digital territory after another, and to maliciously collapse it. Remove the background axiom system within it and turn it into a free 'land' whose title is territory but is actually endless chaos.

That's right, the Undecided Ones appeared suddenly, and the civilization of the Dao Masters didn't know their origins, at least the three Xuan Zhangs didn't know it at all.

Except for this mystery that arouses all kinds of speculation.

Under normal circumstances, the Taoist clan would not pay attention to the actions of the Undecided clan at all. No matter how much unclaimed territory the other party occupies, they don't care.

However, after appearing in the Lost Abyss, the Undecided Clan has deliberately continued to sneak attacks and even attack various 'lands' of the Inevitable Kingdom, plunging it into endless uncertainty and meaninglessness.

This made the Dao Master Clan unbearable.

So after repeated attacks and counterattacks, the eternal war finally broke out between the two super-class clans.

Or it can be said that it was a continuous offensive and defensive war in which the Undetermined Ones mainly attacked and the Dao Masters mainly defended and counterattacked.

Perhaps, this is the biggest reason why this huge territory community is full of hollow and mottled traces of power.

Those traces should be the wounds left here after the Undetermined Ones attacked this territory community.

Now that Mu Cang has become the supreme ruler of this territory, he can clearly feel that it is these undetermined scars mixed with endless uncertainty and meaninglessness that have led to the birth of those endless abnormal things.

--\u003e\u003e

At the same time, from the memories of the three Xuanzhangs, Mu Cang also learned that the number of territory communities like the one he is currently in in the inevitable country established by the Dao Master civilization can be said to be countless and endless.

What is even more shocking is that above the territory clusters, there is an even larger territory aggregation structure - [Boundless Dome Ring].

The number of logical territories contained in those so-called boundless domes are all large bases of various levels and layers.

The overall structure of the "territory" of the inevitable country is built up piecemeal with these boundless domes of various levels and orders as basic units.

And all of this, in a complex form, surrounds an unknown structure called [Omniscient Tower], which exists in a moving or static or non-moving or non-static manner.

As for what the Omniscient Tower contains, whether it contains a large number of logical territories like the boundless dome ring, the three fallen Xuan Zhangs have no idea.

In the final analysis, these three guys are just unknown people in the inevitable country. They have never climbed the tower and have no qualifications to climb the tower. Naturally, they cannot know what its true appearance and essence are.

The three Xuanzhang only knew that the ruler of the Daoist tribe existed in the All-Knowing Tower.

"The All-Knowing Tower where the ruler of the Inevitable Kingdom lives?"

Mu Cang looked up at the other numerical logic territories adjacent to the Godel constructible universe and said indifferently, "After I ascend to the unreachable cardinality, I will go there for a visit."