Many people will definitely be very confused about this.
Why is this so?
Why is the minimum transfinite ordinal w with the same potential as ??, no matter what operation it goes through, whether it is w 1, or w w, or w^w, or w↑w, or even a non-recursive CK ordinal, it still has the same
??Equal potential is still within the jurisdiction of Aleph Zero.
But it is only the power of 2, but it can be equal to the higher uncountable infinity?
In fact, the reason is not as simple as imagined, and it is far from complicated.
First of all, countable infinity is the smallest infinite cardinal number.
Note that it is a cardinal number and not an ordinal number.
Therefore, there is no need to follow the rules of ordinal powers to operate like the smallest infinite ordinal number w.
The operation of ?? needs to follow the rules of base power.
The rules for cardinal powers and ordinal powers are completely different.
According to the operation rules of base powers, the essence of the calculation 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 also be referred to as the 01 function or the Dirichlet function.
This is a kind of domain that is the entire range of real numbers (R), the value range is {0,1}, and it is discontinuous everywhere and is not differentiable everywhere. It is Riemannian inintegrable in any interval, but it is Lebeian in the unit interval.
Lattice integrable measurable functions.
The 01 function, a function set that cannot draw a function image but whose image exists objectively, is completely equal to the power set of the natural number set, and also completely equal to the real number set.
Therefore, according to Cantor's diagonal argument, we can draw the inference result of 2^??>??.
According to the continuum hypothesis (CH), it can be seen that there is no additional infinite base between ?? and ??, so 2^?? greater than ?? is equal to ??.
In fact, among the infinite infinite levels and the finite number of intelligent civilizations with lives in the ten-dimensional universe of the Infinite Seat, no one can know whether 2^?? can be equal to??.
In other words, no one knows 100% whether there is no infinite cardinality between the set of natural numbers and the set of real numbers.
Simply 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 rather than a theorem of an absolute property law.
However, Mu Cang now clearly knows.
There is indeed no additional infinite cardinality between ?? and ??.
At least this is the case in the realm of Dragon Fist. If it is in the realm of Kongyuan or even other stranger realms that may exist, perhaps even the concept of infinite cardinal numbers may not exist.
At the same time, Mu Cang seriously doubted that one of the important reasons why the continuum hypothesis could not be proven or disproven in the ZFC axiom system framework.
This may be caused by the artificially created absolute blank space outside the Aleph Zero Domain, or by the Aleph family's efforts to solidify their power hierarchy and maintain their superior status.
"In short, Alev Zero Domain and Aleph One Domain are closely related. As long as one of them has a big problem, the other one will never be calm."
After saying that, Mu Cang turned his head and looked at the entire countable infinite realm, at the entire endless sea of tribulations, and at all the nirvana dust:
"So, I want to do something to cause chaos in the entire Aleph Zero Domain, so that Mr.?? has to show up and come."
Afterwards, Mu Cang activated [Ling Yue Feifu]:
Activation effect - [I am countable infinity itself ≠ If something goes wrong in the Aleph Zero Domain, it will affect me]
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 now on Mu Cang has become a conceptual entity that is absolutely independent from the Aleph Zero realm and has a countable and infinite self-existence.
In other words, even if the Aleph Zero Domain in the objective world and all the phenomena and rules related to it collapse and disintegrate, they will never harm or even affect Mu Cang in the slightest.
But in turn, with his countless power, Mu Cang was able to subjectively and unilaterally interfere and control the entire Aleph Zero Domain without restriction.
That’s right, it’s such a double standard, that is, both wanting and wanting.
Are you not convinced? If you are not convinced, bear with it.
In short, after buffing himself, Mu Cang started his operation.
"It is not enough to simply use [Ling Yue Fei Fu] to distort the Aleph Zero Domain, because no matter how defiant this ability is, it is still a 'product' that exists within the mathematical logic system of this territory, so
…”
Mu Cang pondered slightly, "We need to add some ingredients to have a two-pronged approach. Let's add...the power of empty numbers."
The so-called foreign monks like to chant sutras.
The T0 skills of this realm plus the mysterious ability of a different realm, one plus one is definitely far greater than two, a perfect match!
Just do it.
Mu Cang immediately activated [Ling Yue Feifu] and [Number of Hollows] at the same time, and they all covered the entire Aleph Zero Domain.
boom!
In an instant, the entire countable infinite realm and everything within it turned into chaos without any reason.
Under Mu Cang's will, all mathematical logic in the entire Aleph Zero Domain began to become extremely hazy.
And the vast distance between finite numbers and infinite numbers, as well as between transfinite ordinal numbers at all levels, which can be called infinite or even infinite and infinite, was also completely confused by Mu Cang until it became blurry and sometimes non-existent.
Including all pure mathematical concepts such as transfinite ordinal numbers and non-recursive ordinal numbers that only exist at the abstract level, they all suddenly became concrete and began to be related to countless entities or virtual phenomena, such as space-time, mass and energy, etc.
The rules and so on merge with each other and merge into one.
The result of these changes is that the entire endless sea of tribulation has completely disappeared.
Instead, there is a vast realm called "Fairy Realm".
The dust of Nirvana that originally spread all over the Sea of Tribulation fled away one after another when the Immortal World was born, falling one by one to the Immortal World and turning into a square of ancient world.
At the same time, the so-called original chapter, the world of truth, the essential world, above the sea of fog, the infinite sea of fog, the seven-dimensional mysterious universe, the torrent of world evolution, the era of the sea, the panorama of the sea, the cluster of singularities...
…and so on, the realms at all levels and levels also distorted and changed, and quickly turned into a series of realms called the mortal world, the true world, the spiritual world, the demon world, the Buddhist world, the Brahma world, the holy world, and the divine world.
, the Yuan Realm, the Heaven Realm, the Martial Realm, the Sword Realm, the Mysterious Realm, the Witch Realm, the Dragon Realm, the Demon Realm, the Ghost Realm... and other smaller realms at all levels and levels.
Then, Mu Cang, who was standing above all of this, took out Yan Qingyang's broken corpse out of thin air, and then said leisurely:
"You unintentionally created the world in which I was born, which means you indirectly created me, so... I will give you a blessing."
As he spoke, this dead body that had lost all its vitality for an unknown number of non-recursive years suddenly blurred into pieces and quickly collapsed into a small ball of light.
As soon as this bright light group was born, bursts of strong vitality and blank thoughts were emitted.
Yes, Mu Cang resurrected Yan Qingyang, but he also sealed all his memories and life essence.
Mu Cang's purpose was to throw Yan Qingyang into reincarnation and be reincarnated into a new world that he had transformed and reshaped based on the entire Aleph Zero Domain.
Let Yan Qingyang embark on the path of evolution again, using his hidden CK ordinal life essence as his golden finger, practice step by step, and finally return to the new endless sea of tribulation... that is, the current non-recursive fairyland.
middle.
Mu Cang believed that with a lot of accumulation, Yan Qingyang would definitely be able to rise quickly from the mundane world, advance all the way to the immortal world, and then reach the pinnacle of the immortal world step by step.
The reason why he has so much confidence 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 you are just a mortal on a small planet in the mortal world, you can't even hit a brick.
As long as you have a strong enough will, a terrifying enough perseverance, a fierce enough talent, a proud enough talent, a smooth enough destiny, a deep enough understanding, and sufficient resources.
He can embark on a long road of evolution, step by step in an almost metaphysical and abstract way, across the chasm between the finite and the infinite, and even fly over the endless layers of infinity and higher-order infinity, and even higher-order infinity.
There is a huge gap in one level, reaching the top and ascending to the fairyland, achieving the non-recursive true immortality.
As the saying goes, the immortal caressed my head and tied my hair to achieve immortality.
Under Mu Cang's in-depth transformation.
From then on, the path of cultivation for the countless intelligent beings in this world is no longer as miserable as it was before, with layers of layers broken and steps leading to an abyss, where only fate favors them in order to advance to the next level of evolution.
Mu Cang is like stroking the heads of all sentient beings, giving all sentient beings the opportunity to achieve transcendence, a road to heaven that everyone can embark on and move forward.
This is His mercy and His boundless love.
As for the reference template for the world view and evolutionary path of the New Aleph Zero Realm, it happens to be Mu Cang's favorites in his previous life - the two fantasy novels "Shading the Sky" and "The Story of a Mortal's Cultivation of Immortality".
“With the continued existence of the new Aleph Zero Domain, its vague and even extremely distorted mathematical logic will inevitably affect the operation of the Aleph One Domain.
And as Mr.??, who is equivalent to the abstract concept of the smallest uncountable infinity, he will definitely feel uncomfortable... under the influence of this bad quality.
So it won’t be long before He will definitely go to the ‘lower realm’ to see what is going on in the Aleph Zero Realm.”
After whispering quietly, Mu Cang turned his head and looked down at the immortal world, and chuckled:
"As for now, let's just relax and watch a show."
Following his gaze, he flew to the boundless sea below the fairy world, and flew into an infinite universe where endless years have passed since his birth.
There is the world where Yan Qingyang was reincarnated.
This world is a vast universe, boundless.
There are countless big stars all over it, all shining brightly.
These stars and starlight are scattered in the vast sky, like a boundless sea of splendid stars.
There is a slightly smaller star, standing majestically on the quiet edge of a vast sea of stars.
This star is called the Earth, and it is the third planet in the solar system. Its territory is so vast that it is unknown how many trillions of miles it is, and it is unknown how many hundreds of billions of years it has passed since its existence.
On its vast surface, there lives an intelligent race called humans.
Countless millions of years have passed since the birth of this civilization.
After such a long period of accumulation, the human race that emerged from the wilderness and ignorance has naturally experienced multiple industrial revolutions along the way and developed into a standard electrical technology type with nuclear weapons, the Internet, space satellites and 5G mobile phones.
Intelligent civilization.
Yan Qingyang was born into this human civilization.
…
No. 33 Middle School, Dai'an City, Longguo, Hedong Province.
In this huge middle school, there are hundreds of teaching buildings.
Each of these teaching buildings towers into the sky and is filled with students of all grades.
In one of the teaching buildings with a height of 2.8 million meters, loud sounds that tore the air and stirred up the wind and clouds were heard in the air at this moment:
“...If you have a small window in your heart, bright sunshine will come in. If you drink some warm stories, there will be a free breeze inviting some flowers or white clouds...
...There are strings in the heart, even if the guest leaves to have a cup of tea, there will still be a ditty slowly playing, and there will still be trees full of sweet-scented osmanthus that turn into the fragrance of wine because of close friends..."
These sounds, which may be green, immature or rough, are exactly the sounds of the students of No. 33 Middle School reciting morning lessons.
After hundreds of hours of morning recitation, the first formal lesson of the day began: history lesson.
After the gray-haired history teacher, Lao Zhang, walked into a classroom with an area of tens of thousands of acres or hundreds of thousands of acres, the 16-year-old female monitor immediately shook her ponytail and stood up with electrified eyes.
He scanned the entire classroom and gave a solemn order:
"Stand up! Say hello!"
Wow——
The tens of thousands of students in the classroom, including Yan Qingyang who was sitting in the corner, suddenly stood up, bowed to the history teacher on the towering podium thousands of meters away, and shouted:
"Hello, teacher!"
Hua Hua Hua——
In an instant, a torrent of air almost like a Category 12 typhoon bombarded the entire podium and Lao Zhang on the stage.
But after enduring this terrifying torrent, Lao Zhang just pushed up his glasses, frowned and shouted:
"The voice is so low, haven't you eaten? Make the voice louder!"
The female squad leader in the audience reacted quickly and immediately gave the order again:
"say hello!"
When the whole class heard this order, they immediately bowed to the distant podium again and shouted:
"Hello teacher!"
boom!!
In an instant, a torrent of breath sound that was dozens or hundreds of times louder than the previous one surged, roaring and boiling, sweeping through the entire classroom and rumbling towards the podium.
Boom boom boom——
Surrounded by bursts of dazzling lightning produced by forcibly ionizing the air, history teacher Lao Zhang nodded with satisfaction and said with a smile:
