Enter the Immortal Cultivation

In the third year of the Yu calendar, Li Zong and Lian Zong reached a new consensus, which was rare.

If a formula has completely consistent connotations in Li Zong's mathematical theory and Lian Zong's mathematical theory, then it can be said that this formula has "absoluteness".

This "absoluteness" undoubtedly gave Li Zong a certain "hope".

For them, this is simply the last salvation and the only gospel that can be found under the ill-considered world-destroying blow.

The existence of "absoluteness" may indicate that mathematical entities are universally present in different mathematical axiom systems.

And if so, this mathematical entity itself may have the property of "actual completeness".

This is their last hope.

Perhaps they need to find a new way to explore the properties of this mathematical entity.

At this point, Feng Luoyi and the Ge Ting School have surprisingly consistent goals.

They even put aside some differences for the time being and explore this field together.

And in this process, Master Hai Ting finally emerged.

Since Lian Zong proved that intuitionistic logic is not safer than the classical logic of the Gotting School, he seemed to have become a different person, silent and taciturn.

And in Li Jing's creation, his autistic tendency became more serious.

However, this did not prevent him from continuing to shine as a mathematician.

Inspired by Su Junyu's research on the continuum, he introduced Feng Luoyi's research results on well-based sets in the infinite axiom and founded a new school - constructivism.

In a certain theory, all entities defined by finite symbols, up to the height of the reflection sequence traversing "the ordinal of all ordinals", are a constructible class.

And the constructible axiom is to declare that the totality of legal sets under the well-based sequence is equal to the "constructible set".

He inherited the idea of ​​"mathematics is a constructed product" of Su Junyu, but accommodated the set theory that Su Jun hated, and completed the preliminary security proof based on Feng Luoyi's well-based set.

Definition is construction, construction is proof, and proof is path order.

It is precisely because of this that he made a small breakthrough in the theory of arithmetic and entered the vision of Qianji Pavilion.

The Goting School was somewhat horrified by this.

The existence of Feng Luoyi and Turing [perhaps Wang Qi can also be counted] made Qianji Pavilion, a branch of Wanfamen, always the back garden of Lizong.

There were also Lianzong monks who walked there, and even the head of Lianzong, such as Suanjun, developed a parallel theory of arithmetic.

However, Master Haiting officially walked into it.

He even had a tendency to show goodwill to Lizong. Master Haiting even proved that the key difference between intuitionism and other logical schools lies in "whether the use of finite symbols can manipulate infinite or even super-finite entities."

But the emergence of Master Haiting is not entirely a bad thing for the basic school.

While Master Haiting rose, he also proposed many things that corresponded to the past theories of Lizong, allowing the Goting School to reflect on itself and discover many things that might not have been discovered in the past.

They found that many of the same mathematical structures are widely present in different axiom systems. The choice of axiom system only affects the number of mathematical structures that can be verified.

The choice and analysis of axioms is actually to determine which basic principles are used as the "starting point" and "benchmark" of mathematics.

The existence of many axioms is not established out of the judgment of the foundation of mathematics, but because the research activities of the monks of Wanfamen need these axioms [such as the definition of addition and the definition of subtraction].

These further strengthened Li Zong's confidence in "mathematical entities".

It was in this context that Su Junyu proposed a cool operation called "transfer model" through the ideas of Master Hai Ting.

If there is a mathematical axiom system A that is consistent in itself, then there is a model of this system.

Adding the axiom "system A is consistent" to the original system, we get "system A is consistent after adding system A". Then, there is "system A is consistent after adding system A is consistent". Then add "system A is consistent after adding system A is consistent" to "system A is consistent after adding system A"... and repeat this way until infinity.

Then, the "consistency" of system A will be transmitted to "infinity" itself, becoming a "countable infinity" property.

This greatly improved Su Junyu, and he almost reached the Refining Void Stage.

Now, he has to suppress his own cultivation like Wang Qi did in the past to adjust his own body.

Under the idea of ​​the transmission model, the "constructive model" was once again mythical.

The operation of the constructible class is "absolute" for any transmission model of mathematical axioms.

Something very rare happened.

With the joint efforts of Lianzong and Lizong, a mathematical result that is valid in both Lizong and Lianzong was established.

So, at this time, the infamy of Hai Ting Zhenren as "Lianzong traitor" even exceeded the sum of the titles of "Lizong traitor" received by Chen Youjia, Feng Luoyi, and Wang Qi.

Even Hai Ting Zhenren himself had to cross the immortal road again and choose to avoid disaster temporarily.

But he didn't care about these things.

He was invincible again.

Just like the infinite axiom proved by Feng Luoyi, under the well-founded set, the proposition that the complete set is equal to the constructible class cannot be proved or disproved.

He was invincible.

The structural faction is already in an invincible position.

Even if Suanjun can slap him to death, he can't change the result.

Feng Luoyi was not happy about this.

He sighed: "When I first agreed to the Geting Sect, I just wanted to rectify the Infinite Axiom and the Law of the Excluded Middle, and rectify the name of Li Zong, so as to offset... the disastrous consequences of Wang Qi's poor calculation. I I really didn’t expect things to turn out like this.”

Wanfamen is completely crazy.

Both Li Zong and Lian Zong have shown their "desperate" tendencies. They no longer care about what will happen to them in the future, they just want to use all their strength to pin their opponents to death in this puddle of mud - even if they themselves are also trapped in the mud.

This is not only the case between Li Zong and Lian Zong, Li Zong is also merciless against Li Zong, and Lian Zong is against Lian Zong.

Master Hai Ting is an example. He obviously has some responders, but he can only withdraw from the sphere of influence of Suanjun.

And Feng Luoyi also felt some malice from the Geting Sect.

On the contrary, Su Junyu was still responding to Ge Ting.

But Feng Luoyi knew that the man named Fan was even trying to "point out" Su Junyu.

He has already detected two or three such malicious "pointers".

However, that bastard who distributed the ball was always just using conspiracy.

I'm afraid he saw something?

Even though Chen Youjia and Su Junyu both showed different tendencies from Wang Qi, they had long-term exchanges with Wang Qi, and their fundamental views on arithmetic were always consistent with Wang Qi's.

They have the same "ultimate goal" and envision the same picture.

Arithmetic is autonomous and will not be influenced by human will.

Therefore, what Fanbach knows will not be changed by the will of Su Junyu or Chen Youjia.

This is really scary.

Master Turing also shook his head: "In this case... the one who looks forward is still burdened with an unknown prophecy..."

Li Zong and Lian Zong, Ge Ting and Shaoli, the two sides are at odds with each other. Although both sides have achieved results, they have also destroyed many of the other's achievements.

While making great progress, both sides have also suffered a loss of vitality.

Under such circumstances, Wang Qi, who is in a foreign country and concentrates on compiling the original calculation, has become one of the few "stars of hope" left in Wan Dharma School.

But it is a pity that... in ten years, no, no, in less than seven years, this "star of hope" is also at risk of falling.

Many people feel that there may be huge loopholes in Wang Qi's calculations.

——This is simply the tune that all schools of dharma will self-destruct.

This desperate thought occurred more than once.

Feng Luoyi was noncommittal.

And in the winter of this year, the Geting Sect announced unhappily that they seemed to have completed further proof of the first question of Ximen's Twenty-Three Questions.

It's more in-depth than Su Junyu's one.

They proposed the concept of "internal model".

Any transitive model that contains "ordinal numbers of all ordinal numbers" is an "inner model".

Constructable classes are the smallest internal models.

Within the constructible class, the cardinality of the infinite set must be in the form of several powers of two. Therefore, the generalized continuum hypothesis holds.

Any equivalent or reduced form of the generalized continuum hypothesis cannot be proven.

After borrowing the definition of Xinlianzong, they finally went one step further.

This scene was so awkward that the cheers inside the singing court were sparse.

They also completed the exploration of the axiom of choice.

At this time, one person showed obvious dissatisfaction.

Su Junyu sighed: "Continuum...its follow-up should not be like this."

The preliminary proof of the continuum hypothesis was completed by him. He also has feelings about this issue.

For him, the internal model was really counterintuitive and contrary to his "aesthetics" as a Christian monk.

how to say……

So unhappy.

"Loops are the most immutable phenomenon in the universe. Deleting loops and refusing to model loops in mathematics is simply..."

He shook his head. Wang Qi is now his boss, and Feng Luoyi is Wang Qi's master. But he still doesn't like the concept of good base gathering... Well, I'm afraid Wang Qi won't like it too much either?

On the other hand, so does Master Hai Ting.

It would be too optimistic to think that there are only "constructible sets" in the totality of set theory.

After all, the entire "internal model" is developed based on the idea of ​​"relative consistency".

Relative consistency itself has the meaning of "compromising and seeking perfection".

Consistency is achieved by discarding those that cannot coexist.

Cut off feet to fit shoes, cut off toes to avoid insects.

He wanted something stronger...