Attributes Skyrocket Infinitely, I Dominate the Multiverse Chapter 640: Twisted Sea of Calamity, the Road to Heaven (1/3)
Theoretically, 2?? is equal to ??.
Many people will be very confused about this.
Why is this the case?
Why is the smallest transfinite ordinal ω, which is equal to ??, still equal to ?? and still within the jurisdiction of Aleph Zero, no matter what operation it is through, whether it is ω1, ωω, or ↑ω, or even non-recursive CK ordinal.
And it is only 2 to the power of ??, but it can be equal to the higher uncountable infinity ???
In fact, the reason is not as simple as imagined, nor as complicated as imagined.
First of all, countable infinity ?? is the smallest infinite cardinality.
Note that it is a cardinality, not an ordinal number.
So ?? does not need to follow the rules of ordinal power to operate like the smallest infinite ordinal ω.
The operation of ?? needs to follow the rules of cardinality power.
The rules of cardinality power and ordinal power are completely different.
According to the operation rules of cardinality power, the essence of the formula 2?? actually represents the number of functions mapped from the natural number set to the {0, 1} function set.
The {0, 1} function can be referred to as the 01 function, or the Dirichlet function.
This is a measurable function with a domain of the entire real number range R, a range of 0, 1, and is discontinuous and non-differentiable everywhere, and is not Riemann integrable in any interval, but is Lebesgue integrable in the unit interval.
The 01 function, a function set that cannot be graphed but has an objectively existing graph, is completely equipotent with the power set of the natural number set and the real number set.
Therefore, according to Cantor's diagonal argument, we can derive the inference that 2??＞??.
According to the continuum hypothesis CH, there is no additional infinite cardinality between ?? and ??, so 2?? greater than ?? is equal to ??.
In fact, in the infinite number of infinite levels and finite number of life and intelligent civilizations in the ten-dimensional universe of the Infinite Quantity, no one can know whether 2?? can be equal to ??.
In other words, no one knows whether there is absolutely no infinite cardinality between the natural number set and the real number set.
Because, in the existing set theory ZFC axiom system, this question or this conjecture hypothesis is neither falsifiable nor verifiable.
This is why the continuum hypothesis is always a hypothesis, not an absolute property law theorem.
However, Mu Cang now knows clearly.
There is indeed no additional infinite cardinality between ?? and ??.
At least this is the case in the Dragon Fist Territory. If it is in the Abyss Territory or even other more unfamiliar territories that may exist, perhaps even the concept of infinite cardinality may not exist.
At the same time, Mu Cang seriously doubts that this is one of the important reasons why the continuum hypothesis cannot be proved or disproven in the framework of the ZFC axiom system.
It may be due to the absolute blank space outside the Aleph Zero Domain, or the Aleph Family's attempt to solidify the power hierarchy and maintain its superior position.
"In short, the Aleph Zero Domain is closely related to the Aleph One Domain. As long as one of them has a big problem, the other will never be calm."
After that, Mu Cang turned his head and looked at the entire countable infinite domain, the entire endless sea of calamity, and all the Nirvana dust:
"So, I want to do something to make the entire Aleph Zero Domain chaotic, so that Mr. ?? has to show up and come."
Then, Mu Cang activated [Ling Yue Feifei]:
Activation effect - [I am countable infinity itself ≠ Aleph Zero Domain will affect me if there is a problem]
Buzz -
In an instant, Mu Cang's form and spirit underwent some extremely mysterious and extremely strange changes.
The result of this change is that from then on, Mu Cang became a self-existent countable infinite concept body that is absolutely independent of the Aleph Zero Domain.
In other words, the Aleph Zero Domain, which belongs to the objective world, and all the phenomena and rules related to it, will never hurt or even affect Mu Cang in the slightest, even if they collapse and disintegrate.
But on the contrary, with the countable and infinite power, Mu Cang can subjectively and unilaterally interfere with and control the entire Aleph Zero Domain without limit.
That's right, it's just that double standard, it's both.
Are you dissatisfied? If you don't accept it, bear it.
In short, after buffing himself, Mu Cang began his operation.
"It's not enough to use [Ling Yue Fei Fei] to distort the Aleph Zero Domain alone, because no matter how amazing this ability is, it is still a 'product' that exists in the mathematical logic system of this territory, so..."
Mu Cang pondered slightly, "We have to add some more ingredients to work on both ends, just add... the power of the void number."
As the saying goes, the monk from outside is good at chanting.
The T0 skills of this territory plus the mysterious power of a foreign territory, one plus one is definitely much greater than two, a perfect match!
Do what you say.
Mu Cang immediately activated [Lingyue Feifei] and [Number of Void] at the same time, and they both covered the entire Aleph Zero Domain.
Boom!
In an instant, the entire countable infinite domain and everything in it became chaotic.
Under Mu Cang's will, all the mathematical logic in the entire Aleph Zero Domain began to become extremely hazy.
And there is no such thing as a gap between finite numbers and infinite numbers, and between various levels of transfinite ordinal numbers.
The vast distance, which was infinite and even infinitely complex, was completely confused by Mu Cang to the point of being unclear and sometimes disappearing.
Including all the pure mathematical concepts such as transfinite ordinals and non-recursive ordinals that only existed at the abstract level, they suddenly became concrete and began to merge with the infinite physical or virtual phenomena, such as time and space, mass and energy, rules, etc., and merged into one.
The result of these changes was that the entire endless sea of calamity disappeared completely.
In its place, there was a vast realm called "Immortal Realm".
The Nirvana dust that was originally spread all over the sea of calamity also fled away at the same time as the birth of the Immortal Realm, and fell one by one under the Immortal Realm and turned into ancient realms.
At the same time, the so-called original chapters, the world of truth, the world of essence, above the fog sea, the infinite fog sea, the seven-dimensional mysterious universe, the torrent of the world, the era of the sea, the panorama of the sea, the singularity cluster... and other levels and levels of the realms and domains that originally existed in these great realms also twisted and changed, and quickly turned into a series of smaller realms called the mortal realm, the true realm, the spiritual realm, the demon realm, the Buddhist realm, the Brahma realm, the holy realm, the divine realm, the elemental realm, the heavenly realm, the martial realm, the sword realm, the mysterious realm, the witch realm, the dragon realm, the demon realm, the ghost realm... and other levels and levels.
Then, Mu Cang, who was suspended above all of this, leisurely took out the broken body of Yan Qingyang out of thin air, and then said leisurely:
"You accidentally created the world where I was born, which is equivalent to indirectly creating me, so... I will give you a good fortune."
As he spoke, this dead body that had long lost all vitality for countless non-recursive years suddenly blurred and fragmented, and quickly collapsed into a small ball of light.
As soon as this brilliant ball of light was born, it spread out waves of strong vitality and blank thoughts.
Yes, Mu Cang resurrected Yan Qingyang, but also sealed all his memories and life essence.
Mu Cang's purpose is to put Yan Qingyang into reincarnation, and reincarnate into the new world that He transformed and reshaped based on the entire Aleph Zero Domain.
Let Yan Qingyang embark on the evolutionary path again, with his hidden CK ordinal life essence as a gifted golden finger, practice step by step, and finally return to the brand new endless sea of calamity... That is, the current non-recursive fairyland.
Mu Cang believes that with great accumulation, Yan Qingyang will definitely rise quickly from the insignificance of the mortal world, advance all the way to the fairyland, and then step by step to the top of the fairyland.
The reason why he is so confident in Yan Qingyang is because Mu Cang has now completely opened up the entire evolutionary path from low to high for countless sentient beings in the Aleph Zero Domain.
From now on, even if he is just a mortal who can't even be blown up by a brick on a small planet in the mortal world.
As long as he has a strong enough will, terrifying enough perseverance, wild enough talent, proud enough talent, smooth enough luck, deep enough perception, and strong enough resources.
He can embark on a long evolutionary road, step by step in a nearly metaphysical and abstract way, step over the natural gap between the finite and the infinite, and even fly over the infinite and high-level infinity and even the infinite and endless layers of huge gaps between the infinite and the higher-level infinity, and even the higher-level infinity, climb to the top and ascend to the fairyland, and achieve the non-recursive true immortal fruit.
The so-called immortal touches my head, and I get immortality after tying my hair.
Under Mu Cang's deep transformation.
From then on, the cultivation path of the countless intelligent beings in this world is no longer the original miserable state of breaking layer by layer and step by step into the abyss, and can only rely on fate to advance to evolution.
Mu Cang, like touching the top of all living beings' heads, gave them the opportunity to live forever and transcend, a road to heaven that anyone can step on and move forward.
This is His mercy, His boundless love.
As for the worldview and evolutionary path reference template of this new Aleph Zero Domain, it happens to be the two fantasy novels that Mu Cang loved in his previous life - "The Great Ruler" and "The Story of a Mortal's Journey to Immortality".
"With the continued existence of the new Aleph Zero Domain, its unclear and even extremely distorted mathematical logic will inevitably affect the operation of the Aleph One Domain.
And as Mr. ??, who is equal to the abstract concept of the smallest uncountable infinity, he will definitely feel uncomfortable under the influence of this evil quality.
So it won't be long before He will definitely go to the "lower world" to check what happened in the Aleph Zero Domain."
After whispering faintly, Mu Cang turned his head and looked down at the immortal and mortal realms, chuckling:
"As for now, just relax and watch the show."
Following His gaze, he flew to the boundless sea of realms below the immortal world, and flew into a boundless universe where endless years have passed since his birth.
There, it is the realm where Yan Qingyang was reincarnated.
This world is vast and boundless.
It is full of countless big stars, shaking and flashing brilliant light.
These stars and starlight are embellished in the vast universe, just like an endless and gorgeous sea of stars.
There is a slightly smaller star, standing majestically in the quiet wilderness of the vast sea of stars.
This star is called Earth, the third planet in the solar system, with a vast area of unknown trillions of miles, and it has been in existence for unknown hundreds of billions of years.
On its vast surface, there lives an intelligent race called humans.
Since the birth of this civilization, it has been unknown how many millions of years.
After such a long period of accumulation, the human race that emerged from the wilderness and ignorance naturally experienced multiple industrial revolutions and developed into a standard electrical technology-based intelligent civilization with nuclear weapons, Internet, space satellites and 5G mobile phones.
Yan Qingyang was born in this human civilization.
…
Longguo, Hedong Province, Dai'an City, No. 33 Middle School.
In this huge middle school, there are hundreds of teaching buildings.
-->>
Each of these teaching buildings is soaring into the sky and filled with students of all grades.
In one of the 2.8 million-meter-high teaching buildings, loud sounds that tear through the air and stir up the clouds are constantly coming out of the air outside the building:
"... If there is a small window in the heart, bright sunshine will come in. If you drink some warm stories, there will be a free breeze inviting some flower fragrance or white clouds...
... If there are strings in the heart, even if the guests leave and the tea is cold, there will still be small songs playing slowly, and there will still be a tree full of sweet-scented osmanthus and turn into wine fragrance..."
These sounds, whether immature, tender, or rough, are exactly the sounds of the students of No. 33 Middle School reciting morning lessons.
After hundreds of hours of morning lessons, the first formal course of the day began: history class.
When the gray-haired history teacher, Lao Zhang, walked into a classroom that was tens of thousands of acres or hundreds of thousands of acres, the sixteen-year-old female monitor immediately swung her ponytail, stood up, and swept her eyes across the classroom with electricity, and ordered majestically:
"Stand up! Say hello!"
Wow——
The countless students in the classroom, including Yan Qingyang who was sitting in the corner, stood up in an instant, bowed to the history teacher on the towering podium thousands of meters away, and said in a low voice:
"Hello, teacher!"
Wow, wow, wow——
In an instant, the air torrent that was almost like a twelve-level typhoon bombarded the entire podium and Lao Zhang on the podium.
But after enduring this terrifying torrent, Lao Zhang only pushed his glasses, frowned and said in a low voice:
"Your voice is so quiet, haven't you eaten? Speak louder!"
The female monitor in the audience reacted quickly and immediately gave another order:
"Say hello!"
When the whole class heard this order, they immediately bowed to the distant podium again and shouted:
"Hello! Teacher! Hello!"
Boom! !
In an instant, a torrent of air sound that was dozens or hundreds of times louder than the previous one, boiling and roaring, swept the entire classroom and rumbled towards the podium.
Boom boom boom——
Surrounded by waves of dazzling lightning produced by forced ionization of the air, history teacher Lao Zhang nodded with satisfaction and smiled:
"Very good, very energetic.
Dear students, please sit down."