Exploring Immortality Cultivation-Chapter 885 - 214: Godel’s Proof Method

"I know! I know!" Su Junyu's excited voice echoed throughout the entire training ground. He shouted frantically, so excited that he shattered the draft papers in front of him with mana, scattering them into the air like celestial maidens scattering flowers.

In recent days, this training ground had been lifeless. A group of arithmeticians whose beliefs were bound to be negated were gathered together, learning and exchanging ideas, much like a group of death row inmates huddling for warmth.

Su Junyu's shout was undoubtedly a huge encouragement to the others.

Bao Xiaofeng's body trembled. He knew what Su Junyu was pondering. Although various signs indicated that perhaps some predecessors had already proven that "incompleteness"—even if not, at least they had refuted the Master's approach.

However, when this result truly appeared before him, he could hardly believe it.

Su Junyu's demeanor also caught the attention of others. In recent days, his peculiar thoughts had already been noticed by many. Hearing about his consideration of "incompleteness," many were watching, some waiting to see a joke, others waiting for his success.

And now, did he succeed?

More people gathered around Su Junyu. Bao Xiaofeng was the first to reluctantly say, "Su Junyu... congratulations... congratulations..."

Su Junyu waved his hand: "It's not time for congratulations just yet. I feel I've inadvertently found a major direction... if successful, there's hope for the Primordial Spirit."

Another cultivator who knew Su Junyu asked, "Primordial Spirit... you've just been in the Golden Core stage for three or four years now, right?"

"The path to the Primordial Spirit. No, if I've truly solved this problem, possibly not only the Primordial Spirit but even Void Refining Nirvana would be certain." Su Junyu's expression was extremely exhilarated: "This could be a Carefree level achievement!"

The crowd gasped in awe.

Many cultivators achieve Carefree level achievements before and after the Primordial Spirit Heaven Barrier, then steadily progress to Carefree. Taiyi Heavenly Venerable is a typical example. Many geniuses who became famous at a young age are like this.

The way people looked at Su Junyu immediately changed. A Carefree level achievement implied an infinitely bright future. Even if one failed while striving in this direction, it would at least refute a wrong path. And in this "negation," the cultivator initiating the challenge could still gain immense benefits.

Moreover, what could be a Carefree level achievement at this time besides the second question among the twenty-three questions—"proof of completeness"?

"You've found a method?" Gao Jiyang hurriedly asked. Although he had already changed his mindset, he still hadn't found a path to incompleteness.

Su Junyu nodded and said, "Yes, but let me make it clear first, you can't take this method and run with it."

"Senior Brother Su, so many people are watching, I don't believe anyone would truly be so shameless." Bao Xiaoya said loudly. She was eager for the incompleteness proof and just waiting to see it.

Bao Xiaofeng also said: "Exactly. Old Su, if anyone tries to steal your idea, I... and my family will support you."

Bao Xiaofeng's "family" included the entire Bo Family, composed of multiple Carefree Practitioners. No one could ignore this warning.

And more people shouted, "The thing might have been proven by a predecessor, so why hide it? Just say it! Come on, tell us!"

These Wanfa Sect disciples had been eagerly awaiting an answer to this question for a long time.

Su Junyu nodded: "Alright, then I'll speak. You're all familiar with the continuum, right?"

"The continuum?" Gao Jiyang immediately sensed something was off: "The second question involves the continuum? Are the first question and the second question two sides of the same issue?"

Su Junyu was also taken aback: "Two sides of one issue? Is there such a saying?"

Then, he realized something as well and couldn't help laughing: "Uh... Sorry, everyone, the inspiration I just had pertains to the first question, not related to the second."

"The first question?" Bao Xiaofeng's eyes widened: "Weren't you thinking about the second question? How did you end up back on the first?"

"A happy accident." Su Junyu laughed: "In order to disprove completeness, I've been thinking about every corner of mathematics, especially within the domain of set theory. Then, I considered that 'axiom of set theory' and found inspiration from it..."

"Hold on!" Gao Jiyang interrupted Su Junyu, then drew a circle in the air with his finger. Nature's spiritual energy gathered into this circle according to his will, forming a mature law. This spirit light circle enveloped those around Su Junyu.

Gao Jiyang said seriously: "We can speculate that the second question has been solved, but there's no indication that the first has been solved by anyone else. Thus, this idea should still belong solely to Junior Brother Su."

"If any nearby senior brothers still want to conduct research in the domain of the first question and wish to maintain their reputation, it's best not to listen any further. Otherwise, if a dispute arises, it won't look good for anyone's reputation."

After a while, two or three Primordial Spirit Stage Grandmasters exited Gao Jiyang's circle. The purpose of this spirit light circle was to isolate internal and external perception. Once outside this circle, no matter what Su Junyu discussed, they wouldn't be able to hear or see it.

If they somehow developed similar inspiration to Su Junyu, it would be impossible to explain logically. Especially since Su Junyu seemed to have a good relationship with the Bo siblings. To leave a poor impression on the imposing Bo Family over this would be a significant loss.

Only then did Gao Jiyang say to Su Junyu: "Junior Brother Su, you can continue. But don't say too much; keep it brief."

Su Junyu sheepishly rubbed his head: "The junior brother thought too much and acted rashly; the senior brother was right to correct me."

"Fortunately, my domain doesn't relate to the first question at all, or I too would be unable to resist listening." Gao Jiyang laughed: "Now, go ahead and speak."

"Tell us!" the crowd urged impatiently.

"Then I'll briefly explain. Firstly, I want to clarify that this is merely a thought, a direction I have currently. I don't have a detailed process nor do I know if the preliminary theories for this process are complete."

"Here's how I think of it. First, we establish a framework. All sets that can exist and hold in accordance with the axioms of set theory are placed within this framework. Then within this framework, we proceed with such operations…"

The area around Su Junyu gradually quieted down.

"Now, I'll think once more about the continuum."

After concluding a correspondence with Lu Xiaoqian, Wang Qi returned to his desk, beginning his contemplation of the continuum hypothesis.

"First, there's Godel's proof on this matter."

"Godel's work involves establishing a massive framework incorporating all 'constructible sets' within the ZF axiomatic system. Within this model, Godel proved that the continuum hypothesis cannot be refuted within the ZF axiomatic system."

"And twenty-three years after Godel accomplished this proof, another mathematician named Cohen produced a proof. He proved that the continuum hypothesis cannot be confirmed within the ZF axiomatic system."

"In other words, the continuum hypothesis is actually an undecidable problem. It exists independently of set theory, whether it's true or false, whether you accept it or not, it doesn't affect anything."

"The famous hypothesis posed based on set theory by its founder is independent of set theory itself, which indeed is nature's caprice," Wang Qi sighed slightly.

"And the forcing method is entirely different from Godel's approach."

"The term 'forcing method' sounds quite aggressive, and in truth, its approach is akin to brute-force cracking. It extends the scope of 'constructible sets' down to the minimum range allowed by the ZF axiomatic system and gradually approaches that result."

Due to this approach involving the 'creation of originally nonexistent sets,' the proof process of Cohen's forcing method is many times more profound than Godel's.

And this is the most crucial part. The significance of the 'forcing method' lies not merely in the continuum. Its emergence allows mathematicians to discover numerous other propositions independent of the ZF axiomatic system.

Purely from a mathematical perspective, the impact of the forcing method is even greater than Godel's Incompleteness Theorem.

In the history of Earth, some mathematicians led by Hilbert ignored Godel until their deaths. A significant part of this was indeed due to a clash of ideas, a struggle of beliefs, as they simply couldn't accept Godel's theories.

However, this also relates to the fact that Godel's argument wasn't "mathematical enough."

To some mathematicians, Godel's Incompleteness Theorem was more of a verbal argument than a mathematical proof process.

"As long as I can complete the forcing method, the mathematics of Shenzhou can reach a new realm."

Wang Qi wrote furiously. For a while, only the sound of the pen rubbing against paper filled the study.

"Searching… searching for a set smaller than the constructible set under the ZF axioms…"

"Independent axioms…"

"Independent…"

In Wang Qi's thoughts, his mana once again underwent subtle changes.

This time, however, no anomalies appeared around him, nor did his mana show any signs of disturbance or collapse.

Everything seemed to change silently. It was as if his mana merely underwent a slight adjustment.

Wang Qi himself didn't know what effect this new adjustment would ultimately have.