Attributes skyrocket infinitely, I dominate the multiverse Chapter 697: Above Choice, Reinhardt (1/3

Mu Cang saw in the long memory of King Peter.

The digital logic territory where the other party was born actually had a master.

All the rules and phenomena, all the logic and domains, were fully controlled by the world cardinality master.

But Peter's [Domination Invasion] was too against the will of heaven.

So when that master realized the existence of Peter and wanted to observe him, everything was irreversible.

In just a moment, the master became Peter's "captive" and became his miserable offspring.

And the possession that led to the complete change of ownership of the territory also made Peter know it suddenly.

So that in addition to everything he knew, there was a wider and more wonderful world.

So, King Peter, who "feeds" on masters and undetermined people... was born.

Since then, no matter which logical territory or weird territory, no matter which master or undetermined person, they all became the hunting targets of King Peter.

Since leaving his "hometown", King Peter has been wantonly taking possession of all the people, objects, and domains he met and saw.

No one has the ability and qualifications to stop him.

Even King Peter eventually became a third-party force independent of the Kingdom of Inevitability and the Kingdom of Freedom.

He was feared by both the Master and the Undetermined.

In a sense, King Peter opened an era that belonged to him alone.

However, after running rampant in all realms and domains of the Lost Abyss for endless years, King Peter's journey of willful behavior was finally ruthlessly interrupted.

That was a long, long time ago.

During that period of time, King Peter had already clearly explored the upper limit of the strength of the entire "world", or the level of the strongest people of the two clans.

That is, Ikaroski level.

As for the higher level, King Peter knew from the intelligence he had obtained that this kind of strong man might only exist in the meaningless source and on the omniscient tower.

In other words, there might be a Reinhardt cardinal level, or even a higher-level cardinal level Xuan Zhang on the omniscient tower, which completely surpassed and surpassed the Ikaroski number.

So, in that ancient era, King Peter, who had already pushed all his masters and undetermined clones to the Ikaroski level and the [Zhizhen Shen level] level, and therefore felt that he was already invincible in the world, killed through the high-level main defense lines with great arrogance and headed for the omniscient tower in the core area of ​​the inevitable country.

And then... there was nothing else.

King Peter now only vaguely remembers that when he led the army of clones composed of Ikarossky levels and stepped into the bottom base area of ​​the omniscient tower... he knew nothing.

When he woke up again, he could only find some unfamiliar concepts such as "Eye of the Supreme", "Crossing Choice", "Above the Mysterious Palm" and so on in his extremely vague memory.

At the same time, King Peter also found that most of his clones had perished.

The few remaining clones also fell from the Ikarossky level to the "low" level of the world base, unreachable base, Malo base, etc.

And what's even more outrageous is that King Peter found that his [Dominate Invasion] ability was also affected by an unknown influence.

Whether it is the target of possession or the number of possessions, they are inexplicably sealed in the scope of the super huge base field, and there is no possibility of unlocking.

Regarding this point, in the following years, King Peter had used clones of various levels to conduct various tests.

But, all failed.

Whether facing the Ikaroski-level Xuan Zhang or the top Undetermined of the [Zhizhen Shen] level, his clones could not carry out any possession and could only be ravaged and killed by the other party.

In the end, except for the destruction of some clones in vain, King Peter could not gain anything.

At the same time, King Peter also discovered a situation that shocked him.

That is, all the "traces" left by him inside and outside the entire Kingdom of Inevitability and the Kingdom of Freedom have become blurred or even unrecognizable.

Many masters and Undetermined who knew more or less information about King Peter had forgotten his existence.

The main defense lines, branch defense lines, dome rings, boundless dome rings, and territory communities of all levels and levels that were broken, broken, disintegrated, and even destroyed and disappeared because of King Peter were all distorted and mutated.

Some of them changed from huge main defense line structures to smaller boundless domes, or from dome rings of the kingdom of necessity to the bizarre territory of the kingdom of freedom, or returned from the end that had long been destroyed, or even never appeared at all.

In short, it was as if King Peter had never existed before, and all the changes related to him had become vague and specious rumors.

Faced with this extremely strange situation, King Peter was completely frightened.

He knew that he must have angered an unknown supreme being, which led to this horrifying result.

And that supreme being was likely to exist on the Tower of Omniscience.

So from then on, King Pitt changed his previous arrogant and domineering style of behavior and became much more low-key and secretive.

Although I don’t know why the Supreme One didn’t take my own life, He who was timid and afraid of death still hid--\u003e\u003e

He hid for a long time before he dared to show up.

And after appearing, he didn't dare to seize the body at will, let alone occupy the territory and become the king at will. He only dared to secretly disperse all the clones in countless territories.

However, although he lost the courage to continue to do things, driven by curiosity, King Peter still secretly began to investigate the "Supreme Eye", "Transcendence Selection", "Mysterious" in the long years that followed. "On the palm of your hand"...etc. information on vocabulary concepts.

Perhaps because he was patient and persistent enough, King Peter actually found out a thing or two about these unfamiliar words.

The first is the "Supreme Eye", which can be known based on the information he spent time and effort gathering from all over the realm of necessity and freedom.

The so-called "Supreme Eye" is an unconfirmed and mysterious legend with a long history.

Legend has it... that the entire "world" where all the masters and undecided ones live are born from the "Supreme Eye".

At the same time, after the birth of the "world", the [meaningless source] was born from the "Supreme Eye", and finally the undecided family and the free country emerged.

"Crossover Choice" and "Above the Mysterious Palm", after the exploration and research of Peter King, it was finally discovered that there is a profound connection between the two vocabulary concepts. They are also two concepts that are spread in various fields. Mysterious rumors.

The former, the so-called "overpass selection", actually refers to... if you can jump over or bypass the axiom of choice, you can reach a new field.

The latter refers to exactly that new realm - above the Mysterious Palm.

Yes, Xuanzhang is not the end of the Taoist's evolutionary system.

Above it, there may be a grand and majestic new realm.

As for the content of the axiom of choice, given a series of sets, one element can be selected from each set, which is an important part of the ZFC axiom system.

Without it, the measure theory about real numbers will fall apart, and there will be a series of mathematical structures of uncountable infinity and above, that is, the zero and zero large cardinal numbers and high-order large cardinal numbers, and various large and small occurrences will also occur. changes.

In short, through the collection of a large amount of intelligence and careful research, King Peter finally concluded.

If you want to truly and safely set foot on the omniscient tower, even if you just climb to the base area at the bottom of the tower.

It is very likely that the life and strength level of the climber is required to reach the corresponding level of the large base after completely overcoming the axiom of choice and entering a new field.

That is, the Reinhardt cardinality, and the higher-order large cardinality above it.

Yes, it is that big cardinal number that is always entangled with the concept of "01".

The reason why it always has the name "01" is closely related to the axiom of choice.

In other words, it is because of the contradiction and incompatibility with the axiom of choice that the Reinhart cardinality is labeled "01".

The so-called Reinhart cardinality is an important mathematical concept in set theory.

Its definition and structure can be elaborated from many aspects.

First of all, the definition of Reinhart cardinality is a special type of cardinality that exists under the set theory system ZF axiom system without the axiom of choice (AC).

Expressed in mathematical language, there is a non-trivial elementary embedding j: V→V, crtjκ, and this κ is the Reinhart base number.

So specifically, it refers to the critical point crtjκ of this non-trivial basic embedding.

The embedding j is elementary, which means that all true propositions that are true before embedding will still be true after embedding.

The other V refers to the entire von Neumann universe of set theory, that is, the true type of all sets.

Therefore, if we combine these and elaborate further, Reinhardt cardinality involves a non-trivial basic embedding, which maps the entire class V of set theory to itself and has specific critical points.

Among them, there is also a characteristic of Reinhardt base numbers - self-embeddedness, elementary embedding from itself to itself.

The so-called "non-trivial embedding" in the previous paragraph refers to the Reinhart cardinality itself, which is actually the critical point of the basic embedding.

As for the expression of this critical point in mathematical language, it is... κ is the critical point of embedded j, that is, for all ordinal numbers α less than κ, there is jαα, but jκ≠κ.

This embedding then maps the entire class V of set theory to itself, and it is not an identity mapping—that is, there is some set x such that jx≠x.

At the same time, since embedding j has a critical point κ, this means that for all ordinal numbers α less than κ, there will be jαα, and for κ itself, there will be jκ\u003eκ.

To put it more specifically, this embedding will have specific properties, which will map certain elements in V to other elements in V, and some structures or properties of the set will remain unchanged during the mapping process.

Secondly, since it cannot be described or defined by first-order logic language, Reinhardt cardinal numbers are also undefinable.

In addition, apart from these, what really causes the Reinhardt base number to have the title property of "01" is the inconsistency between it and the standard set theory axiom system ZFC with the axiom of choice.

It can also be called the library-energy inconsistency theorem.

The content of this theorem is that in the set theory system with the axiom of choice, there is no non-trivial basic embedding that can map the entire class V to itself.

To put it more specifically, within the overall framework of the ZFC system, there is no cardinality that can satisfy the definition of Reinhardt cardinality. It must be established in a set theory system without an axiom of choice (such as the ZF system). and discussion.

The reason for this is because the definition of Reinhardt cardinality involves non-triviality--\u003e\u003e

Basic embedding.

According to the library energy inconsistency theorem, this embedding cannot be established in the ZFC axiom system at all, or it will lead to serious inconsistencies, which will lead to various conclusions that are contradictory to known mathematical facts.

In addition to this theorem, there are other mathematical results and reasoning that show that Reinhardt's cardinal number and the axiom of choice cannot coexist logically. These counterexamples further support the incompatibility between the two.

Therefore, in a self-contradictory axiom system Reinhardt Cardinality ZFC, naturally any messy proposition can be given a truth-forcing proof.

For example...01.

Therefore, the Reinhardt base number has no choice but to have the so-called "01" label.

In fact, it is not only the Reinhardt cardinality that is inconsistent with the axiom of choice and the ZFC axiom system.

The same is true for the Berkeley cardinal numbers, super Reinhardt cardinal numbers, unbounded closed Berkeley cardinal numbers, and even the larger and more distant known and unknown large cardinal numbers above it.

The further essential reason why such contradictions arise is because the addition of the axiom of choice provides set theory with too many degrees of "choice" freedom.

For this difficult problem, either accept the ZF axiom of the existence of Reinhardt cardinal numbers and do not choose the axiom; or accept the ZF axiom of selection and do not want the axiom of the existence of Reinhardt cardinal numbers; or... build an axiom system more powerful than ZFC.

This expanded and upgraded higher-order axiom system may include some additional axioms that allow the existence of Reinhardt cardinal numbers, and then allow the establishment and existence of Reinhardt cardinal numbers and more powerful cardinal numbers above it.

"So the so-called omniscient tower..."

Mu Cang, who had exhausted all the memories of King Pit, slowly turned his head and "looked" at the empty abyss of loss, as if he was "looking" at the omniscient tower located somewhere, and said quietly:

"Could it be a higher-order axiom system... that can accommodate the existence of Reinhardt's cardinal logic configuration?"