Enter the Immortal Cultivation

It was Feng Luoyi who fired the first shot in this world of great contention.

Although Go Ting Zhai had been handed over to He Wail, the first disciple of the Master of Calculation, who was a Lian Zong monk, the Go Ting School was still the orthodox of the Li Zong and the direct line of the Master of Calculation. This point would never be changed simply because of He Wail or any other person.

Perhaps a hundred years later, Go Ting Zhai would eventually become something else, but He Wail alone could not turn this stone.

The core monks of the Go Ting School were already blood-thirsty, and they were deliberately trying to equate the Lian Zong Calculation with the part of the Li Zong Calculation that was defeated by the Unconstrained Calculation, and put them in the attack range of the Unconstrained Calculation.

But the first to achieve the results was Feng Luoyi, the genius known as "inhuman".

It should be said that Feng Luoyi found a brand new idea.

They claimed that the ideas before set theory were all problematic.

We should not start from "all", but from "nothing".

All "sets" must be constructed from the "empty set".

In other words, only sets constructed from the empty set are recognized as legal sets.

All other sets are problematic, and are empty castles whose foundations have been removed by inconsiderate calculations.

Whether it is a finite set or an infinite set, it must start from the "empty set".

The empty set ? corresponds to 0, {?} corresponds to 1, and {?, {?}} corresponds to 2. If all sets, including infinite sets, have similar good order, then induction beyond infinity can be implemented - just like ordinary mathematical induction.

Then, all the axioms of the "heavenly principle system" [ZF axiom system], which is the highest achievement of Li Zong, can be realized in the well-based set.

This is Feng Luoyi's proposition.

This genius completed this great argument in two papers.

Any proof construction must be of finite length, and the proof of contradiction is no exception. The infinite axiom - the axiom of the existence of infinite sets of natural numbers, only uses the successor operation and the empty set operation. These two operations have corresponding principles in the Lianzong arithmetic. Therefore, these two arithmetic principles are absolute between the Lianzong arithmetic and the Lizong arithmetic. In other words, the Lizong arithmetic and the Lianzong arithmetic actually have a considerable degree of consistent connotation.

This is the "absoluteness" of the two arithmetic principles.

Therefore, if the axiom of infinity has a contradiction, then this contradiction will also appear in the arithmetic through a "finite" translation process.

The infinite function axiom is safe.

As soon as this paper came out, the Lianzong monks vomited blood on a large scale.

Everyone knows that the Lianzong, especially the Shaoli School represented by the Lianzong in modern times, denies "infinity" and "the law of the excluded middle". The Calculator believes that there are no infinite objects in the material world, and there should be no infinite objects in the world of mathematics.

This shook the foundation of the Lianzong.

Countless Lianzong mathematicians scratched their heads and wished they could write a paper immediately to fight back against Feng Luoyi.

However, soon, Feng Luoyi's second paper put an end to all the arguments.

"If the negation of the axiom of infinity is taken as an axiom, the contradictions in the finite order will be more conveniently reflected in other axioms."

"Therefore, in a sense, the axiom of infinity cannot be proved or disproved."

This was like a bolt from the blue, and all the Lianzong mathematicians were speechless.

Generally speaking, "falsifiability" is the foundation of the modern method of immortality. Things that are not falsifiable are not worth discussing.

However, the status of mathematics is slightly special.

Even the mathematicians themselves cannot clearly say whether their work is "discovery" or "invention".

On this point, Suanjun and Wang Qi absolutely hold completely opposite views.

Of course, from the level of Meishen, this dispute is a complete joke.

Wang Qi also had this tendency after encountering Meishen.

He even said in the preface of formal linguistics that this debate is purely the chaos of natural language, and it is necessary to distinguish between "discovery" and "invention", two completely different concepts.

But no matter what, in the field of mathematics, a theory that cannot be proved or falsified is allowed to exist.

But it is like theology, which is self-contained in its own logic.

Even if you want to destroy it, it is difficult to do so.

For ordinary people, this is an unknown area where "you don't know what is inside".

But Feng Luoyi is clever in that he directly proved another point at the beginning.

The infinite axiom is safe.

"I don't know what is inside", but it is "safe".

This also shows from the side that Lian Zong's criticism of the concept of "infinity" is really meaningless.

And Chen Youjia's paper was released soon after.

This made all the Li Zong monks uncomfortable for a while.

Some people even cursed on the spot: "Traitor!"

Li Zong traitor!

In the past, almost all monks took the Ji faction for granted as Li Zong. Wang Qi also clearly expressed his position of Li Zong.

And now, Chen Youjia, a Li Zong monk, published a paper with obvious Lian Zong tendencies.

How could they not be angry? How could they not be angry?

But even angrier than them were the many grassroots disciples of Qianji Pavilion.

Type theory directly deleted Turing completeness and loops.

Turing himself found it quite interesting, and even assigned it as a topic.

But when the disciples of Qianji Pavilion found this out, they immediately became excited.

For them, a calculator system without Turing completeness and loops was unimaginable.

They felt that their holy domain had been violated.

The "free flight" of the past has now become "dancing with shackles".

The voice of Chen Youjia "traitor" also caused a stir in Qianji Pavilion.

In this regard, Turing Zhenren could only smile bitterly: "This is... It seems that I am sorry for that girl."

Feng Luoyi shook his head: "Let them make trouble. For these guys who have no intention of calculating, what effect that girl's theory has, they will always know in the future."

"Are you talking about incorporating 'type theory' into 'Turing completeness'?" Turing Zhenren smiled: "Wo has thought about it these days, and it may cause vicious disharmony."

"But Wang Qi told me that this can greatly lower the threshold of the calculator technique." Feng Luoyi said.

With the promotion of the Wanxiang Guawen, he still trusts his disciples' vision in this regard.

"I'm afraid that those who don't know these two things will make mistakes at the critical moment." Turing Zhenren hesitated. The language associated with type theory will shut down directly if there is a slight mistake, which is considered safe. Although the methods used in the past will output unknown results, it is because the cultivators are already familiar with this field, so they rarely fail at critical moments.

But if the two are really combined...

What if the calculator crashes at the critical moment?

"If it is the main magic weapon, it is naturally another matter, but it does have advantages when practicing." Feng Luoyi said.

Master Turing suddenly laughed and said, "Brother Yuehan now looks like he has nothing to do with it. Don't you know? This wave of "traitors to the Li Sect" has also blown to you, right?"

Feng Luoyi lowered his eyelids: "They are just stubborn elements... old things who are not resilient. A careless calculation did not let them see the truth clearly, so they are like this."

In fact, Feng Luoyi's thesis was not without opposition within the Li Sect.

It was because of the opposition of the monks of Qianji Pavilion to Chen Youjia that another wave of opposition was triggered.

Some monks of the Li Sect thought that Feng Luoyi was crazy.

He actually eliminated the concept of "cycle"!

Cycle is a concept that exists widely in nature. The spiritual power cycle represented by the breathing of heaven and earth, the atmospheric circulation, and the water cycle are all a kind of cycle.

So, something like cycle should exist in mathematics-in fact, it is even a key part of the Wanfamen technique.

"Circulation cannot be eliminated from logic."

Those Lizong monks said so.

Chen Youjia and Feng Luoyi both took the "self-fingering" of the "two paths" of the Master of Calculation.

Even the two of them inspired each other to a certain extent.

It's just that from the basic point of view, Feng Luoyi is the most orthodox Lizong, while Chen Youjia has accepted a lot of Lianzong's ideas from the Master of Calculation.

But they are pointing in the same direction.

Feng Luoyi never cares about this little thing. He said to Master Turing: "It's enough that there is no opposition within the Go Ting Sect. Next, it depends on them, right?"

Master Turing did not comment. Everyone knows that the Go Ting Sect is already in a desperate situation at this moment.

Under the crushing of the unconsidered calculation, they can't care about much.

However, Master Turing did not expect that a few days later, the Go Ting Sect's last-ditch attack would come so fiercely.

In fact, it was Feng Luoyi's frequent conversion of Lianzong and Lizong's calculations that attracted their attention.

From this, they extracted an unnamed function at the logical level.

"Double negation translation". [It is called "Gödel-Genzen" translation on Earth]

"Assuming that the law of the excluded middle is contradictory, a negative proof of this assumption can be constructed."

"And the law of the excluded middle can be equivalent to the removal of double negation."

These are two topics that already have simple proofs.

At this time, the Gotting School proposed a further idea.

"If double negation is added to the proof in classical logic, can this proof be equivalent to the constructive proof of intuitionism?"

In other words, if the part of the Gotting School's arithmetic that belongs to the "law of the excluded middle" is removed, then is this thing equivalent to the constructive proof of the Shaoli School?

The answer is, "yes".

It uses the arithmetic of the chain of opposites to determine the law of the excluded middle opposed by the chain of opposites, which is also safe.

As safe as the axiom of infinity.

In other words...

The arithmetic of the chain of opposites is not much safer than the arithmetic of the chain of opposites.