Into the Immortal Cultivation - Chapter 2092

[Chapter 105 The Last Scene [Second Update]]

In the third year of Yuli, Lizong and Lianzong rarely reached a new consensus.

If a formula has completely consistent inner meanings in the Lizong Arithmetic and Lianzong Arithmetic, then it can be said that this formula is “absolute”.

This “absoluteness” undoubtedly gave Lizong some kind of “hope”.

For them, this is simply the last redemption and the only gospel that they can find under the ill-conceived blow of the world.

The existence of “absoluteness” may just indicate that mathematical entities are ubiquitous in different mathematical axiom systems.

And if this is the case, the mathematical entity itself may have the property of “actual completeness”.

This is their last hope.

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

At this point, Feng Luoyi and the Ge Ting faction are surprisingly consistent.

They even put aside some differences for a while to explore the field together.

And in the process, Hai Ting finally came to the fore.

Since Lian Zong proved that intuitionistic logic is no safer than the classical logic of the Getin School, he seems to have changed himself, silent and reticent.

In Li Jing’s initiative, his autistic tendency is even 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 of Liangji set in the infinite axiom, and created a new school of structuralism.

In a certain theory, all entities defined by finite symbols, up to the height of the reflection sequence traversing “the ordinal of all ordinal numbers”, is a constructible class.

The constructibility axiom is to declare that the totality formed by the legal sets under the well-founded sequence is equal to the “constructibility set”.

He inherited Jun Jun’s thought that “arithmetic science is a constructed product”, but accommodated the set theory that Jun Jun disliked, and completed a preliminary security proof on the basis of Feng Luoyi Liangji’s set.

Definition is construction, construction is proof, and proof is road rank.

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

The song court party was a little terrified by this.

The existence of Feng Luoyi and Turing (maybe Wang Qi can also be counted) makes Qianji Pavilion, a branch of Wanfamen, always the back garden of Lizong.

There have also been cultivators Lianzong who have walked there, and even the head of Lianzong, such as Shunjun, has developed a parallel calculator theory.

However, Hai Ting has officially entered it.

He even has a tendency to show his favor to Lizong. Hai Ting even proved that the key difference between intuitionism and other logic schools lies in “whether infinite or even transfinite entities can be manipulated by using finite symbols.”

But the appearance of Hai Ting’s real person is not entirely a bad thing for Ji Pai.

At the same time as the rise of Hai Ting, he also proposed many things corresponding to the past theory of Lizong, which enabled the Ge Ting faction to reflect on itself and discover many things that may not have been discovered in the past.

They found that many of the same mathematical structures are widespread across different axiomatic systems. The choice of the axiomatic system only affects the number of mathematical structures that can be demonstrated.

The selection and analysis of axioms is actually to determine which basic principles should be used as the “starting point” and “benchmark” of mathematics.

The existence of numerous postulates is not established out of judging the foundation of arithmetic, but is established by the research activities of the cultivators of Wanfamen who need these postulates [such as the definition of addition and the definition of subtraction].

These further strengthened Lizong’s confidence in the “numerical entity”.

It is against this background that Su Junyu, through the idea of Hai Ting real person, proposed a show operation called “transfer model”.

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

Adding the axiom of “system A is consistent” into the original system, we get “system A is consistent after adding system A”. Then, there is “system A is consistent and the system after adding system A is consistent”. Then add “system A is consistent after adding system A is consistent”, add “system A is consistent after adding system A”… Repeat this until infinity.

Then, the “consistency” of system A will be passed on to “infinity” itself, becoming a “countable infinity” property.

This made Su Junyu a great improvement, and even almost reached the stage of refining.

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

And under the ideology of the transfer model, the “constructive model” is once again mythological.

The operations of constructible classes are “absolute” with respect to any transfer model of the axioms of arithmetic.

A very rare thing happened.

With the joint efforts of Lianzong and Lizong, a mathematical achievement established within Lizong and Lianzong was established.

Therefore, at this time, the infamy of Hai Ting’s “traitor of Lianzong” even exceeded the sum of the titles of “traitor of Lizong” received by Chen Yujia, Feng Luoyi, and Wang Qi.

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

But he doesn’t care about that anymore.

He was invincible again.

Like the infinity axiom proved by Feng Luoyi, under a well-founded set, the proposition that the complete set is equal to a constructible class cannot be proved.

He is invincible.

Structuralists are already invincible.

Even if you can slap him to death with a slap, you can’t change the result.

For this, Feng Luoyi was not happy.

He sighed: “When I first agreed to the Ge Ting School, I just thought that to rectify the name of the Infinity Axiom and the Law of Excluded Middle, and to rectify the name of Li Zong, it can be considered to offset the catastrophic consequences of Wang Qi’s ill-conceived calculation. I I really didn’t expect that things would turn out like this.”

Wanfamen went completely crazy.

Both Li Zong and Lian Zong have shown a tendency to “despite everything”. They don’t care about their future, they just want to do their best to kill their opponents in this pool of mud even if they are also deeply mired in the mud.

Not only between Li Zong and Lian Zong, but Li Zong to Li Zong and Lian Zong to Lian Zong, they are equally ruthless.

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

And Feng Luoyi also felt a little malicious from the Ge Ting faction.

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

But Feng Luoyi knew that the surnamed Fan was even trying to “point” Su Junyu.

He has already detected two or three such malicious “points”.

However, the bastard who scored the ball has always just used conspiracy.

I’m afraid he saw something?

Even though Chen Yujia and Su Junyu both showed different tendencies from Wang Qi, they had communicated with Wang Qi for a long time, and their fundamental views on arithmetic were always the same as Wang Qi.

They have the exact same “ultimate goal” and dream of the same vision.

Algorithm is self-contained and will not be transferred by human will.

Therefore, what Fanbach knows will not be changed by Su Junyu or Chen Yujia’s will.

This is really scary.

Real Turing also shook his head: “In this case… the one who looks forward is also burdened with an ominous prophecy…”

Li Zong and Lian Zong, Ge Ting and Shaoli, the two sides are considered to be in conflict with each other. Although there are achievements between each other, they also dismantle each other’s achievements.

While making great progress, both sides are also hurt.

Under such circumstances, Wang Qi, who was in a foreign country and devoted himself to editing the original calculations, became the “star of hope” left in the current ten thousand methods.

But it is a pity… Ten years, no, no, in less than seven years, this “Star of Hope” is also at risk of falling.

Many people think that there may be huge loopholes in Wang Qi’s arithmetic.

This is simply the tune of Wan Dharma’s self-destruction.

This kind of desperate idea has appeared more than once.

Feng Luoyi was noncommittal.

And in the winter of this year, the Gothing faction declared unhappily that he seemed to have completed the further proof of the first question of the twenty-three questions of Jimen.

It’s more in-depth than Su Junyu’s.

They came up with the concept of an “internal model”.

Any transitive model that includes “ordinal of all ordinals” is an “internal model”.

Constructable classes, the smallest inner model.

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

And any equivalent or reduced form of the generalized continuum hypothesis is unprovable.

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

This scene is so awkward that the cheers inside the song court are sparse.

They also completed the exploration of the axiom of choice by the way.

At this time, one person showed obvious dissatisfaction.

Su Junyu sighed: “Continuum… Its follow-up shouldn’t be like this.”

He completed the preliminary proof of the continuum hypothesis. He also has feelings for this issue.

For him, the inner model is really against intuitive feeling, against the “aesthetics” he has as a monk of the fundamentalism.

What can I say…

it’s so frustrating.

“Cycles are the most unchanging phenomenon in the universe. To delete them in mathematics, to refuse to model them, 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 collection… Well, I’m afraid Wang Qi won’t like it too much?

On the other hand, so is Hai Ting.

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

In the final analysis, the whole “internal model” was developed from the idea of “relative consistency”.

Relative consistency itself has the meaning of “compromising”.

Consistency is achieved by letting go of those that cannot coexist.

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

He wants something stronger…

[End of this Chapter]