Exploring Immortality Cultivation-Chapter 888 - 217: Sticking a Foot In

In an incense burner, wisps of blue smoke swirled. In the room, there was no visible source of light, yet some unknown light illuminated every inch of the space. High-grade large sheets of Xuan paper were neatly arranged on the desk. The owner of the study preferred soft brushes, so there was also an inkstone on the table. The ink in the stone was even and viscous, resembling both oil and water. A brush lay nearby.

That ink seemed to be waiting, waiting for the moment it would turn into a world-shattering article.

But in reality, the ink seemed freshly ground, merely the spell effect intrinsic to the inkstone. The brush had long remained untouched.

The owner of the ink and brush had been frozen here for a long time.

Xi Baiche, Master of Calculation, no matter what words he wrote, would be revered as masterpieces outside. Though his writing was indeed excellent, it was his identity that was truly prestigious.

However, even such a divine person had a day of furrowed brows.

"Alas... still no progress." He shook his head, the serene and distant gaze barely concealing his anxiety.

These days, he had poured everything into his efforts. There had been results, but they were negligible. Compared to his own grand ambitions, they were trivial.

"Teacher..." One of Xi Baiche's disciples looked at him with concern from behind: "You seem overly worried lately..."

"Yes." The Master of Calculation sighed deeply: "The Mathematics Lord... speaking purely on the Mathematics Dao, there's no equal to him. If such a person attacks, I wouldn't be myself if I weren't afraid."

The Master of Calculation was not innately a genius. Among his peers, he was considered a "hardworking genius," diligently cultivating himself with persistence. Naturally, the geniuses among his peers had already fallen behind him.

In his life, he had neither been unbeatable nor without fear. Therefore, he was not ashamed to discuss these.

Naturally, he spoke his mind.

"Teacher, perhaps you should take a break as you usually do, relax a bit. The completeness barrier is not necessarily something you must overcome. You see, three or five years ago, we already proved the completeness of many branch systems in mathematics..."

"He Ke." Xi Baiche called the student's name directly: "Have you heard any news recently?"

This Half-step Leisure cultivator was one of his early disciples, and naturally, the relationship with the Master of Calculation's later disciples was different. He dared not look at his teacher's back, lowering his head: "Recently... I've had contact with Junior Brother He..."

He Wai'er was among the accomplished disciples of the Master of Calculation. Many of the Master's disciples referred to him as Senior Brother, but the elder ones like He Ke called him Junior Brother.

Xi Baiche showed a headache: "Wai'er... that guy changed after believing in Lianzong. What did he say?"

"I probed his intentions, and things do seem really unfavorable..."

"It seems the Mathematics Lord still has some skills." The Master of Calculation nodded, unconcerned: "There are many flaws in my theories; if he could point out one or two, I'd be grateful. Just that he probably disdains thinking about this domain, huh."

He Ke gritted his teeth: "And also, Mr. Feng... Mr. Feng knows too. Only, they all worry you might not endure..."

"I wondered why my old friend has been acting so secretly recently, so this is it?" Xi Baiche's expression grew solemn: "It seems it's really a major problem. How fundamental could it be? Where exactly am I..."

He Ke felt a chill inside: To this point, the teacher's thoughts haven't leaned towards "Completeness"... he doesn't doubt the completeness of mathematics at all...

"Teacher, we still have a fallback. We can still prove the completeness of the system from outside the system using Transfinite Induction..."

He Ke didn't speak for long. The Master's eyes stared at him. Those black eyes seemed to possess boundless might. Soon, he couldn't continue. After a long while, the Master finally pulled his gaze back to the proof on which he had long stalled. He seemed to want to trace this proof and "see" the other shore of mathematics.

"We must know, we shall know..."

He murmured softly.

"All in all, this is my recent thought process." Wang Qi reported his thoughts to Feng Luoyi with a calm expression.

So-called "thought processes" sometimes require concealment, yet sometimes it matters not to speak them aloud.

To Feng Luoyi, Wang Qi's thought process was such.

Feng Luoyi was already a pinnacle cultivator. There were no higher demands on life or profit for him. What this Great Mathematician sought was nothing more than the Great Dao and to get rid of the so-called "Ultimate Picture." Beyond that, no interest could sway him.

In fact, for the realization of his ideals, the more talented juniors like Wang Qi, the better. From this angle, Feng Luoyi's support for Wang Qi was indeed unreserved.

This is also why those who have already "reached the top" are not stingy in helping the younger generation.

Feng Luoyi pondered Wang Qi's approach in his mind. He thought: "Judging by your momentum, it seems you're determined to break through these twenty-three problems."

"No, no." Wang Qi hesitated. How terrifying are the twenty-three questions? Even on Earth, several of these twenty-three questions remain unresolved. Even if he is confident in independently solving one or two of them, he wouldn't dare claim he can solve them all.

Feng Luoyi shook his head: "Who knows what the future holds? Five or six years ago, I didn't see that you could follow that Completeness to break through the second question, the tenth question—hmm, perhaps you'll also add the first question." 𝑓𝘳𝘦𝑒𝑤𝑒𝘣𝘯ℴ𝘷𝘦𝓁.𝑐𝑜𝑚

"But..." Feng Luoyi's gaze suddenly sharpened: "You seem to always like to take detours."

"Detours?"

Wang Qi was bewildered. The Forcing Method is the perfect choice for solving the Continuum Hypothesis problem, isn't it? Why is it a detour?

"Recently, I've seen a thought, very peculiar, very interesting." Feng Luoyi stared into Wang Qi's eyes and said: "Create a system by unifying all sets that can be established, comparing the sizes of infinity within infinity..."

Wang Qi was stunned: "Someone thought of it?"

This was exactly the idea that Godel used to prove the Continuum Hypothesis!

Within the confines permitted by ZF axioms, deduce and prove that if the ZF axiomatical system is consistent, then the Continuum Hypothesis is true. Within the ZF axiom system, the Continuum Hypothesis cannot be disproven.

For the twenty-three questions, this should already count as a perfect answer.

However, "cannot be disproven" does not mean "confirmed."

Following this idea, one can also prove that within the ZF axiom system, the Continuum Hypothesis cannot be confirmed.

In other words, within the realm of set theory, the Continuum problem is an undecidable problem. If this "neither confirmed nor disproven" conclusion came earlier, there would be no need to strictly prove the tenth question, as it would serve as an absolute counterexample to "decidability."

It can only be proven outside the existing axiom system.

The Forcing Method is about breaking free from the constraints of the original axiom system, creating one's own system and determining one's own path.

Yet, if set theory and the existing axiom systems are the cornerstones, then how should this "self-created" system be evaluated? Outside the cornerstone? Within the mathematics domain?

This also contributes to the significant meaning of the Forcing Method.

Upon hearing Wang Qi's exclamation, Feng Luoyi's eyes flashed: "You've thought of this approach too, didn't you?"

"Vaguely thought of it, but... I don't really like it." Wang Qi could only say this. In terms of the simplicity and fluency of the theoretical proof process, Cohen's Forcing Method is far inferior to Godel's thinking. Unless one is silly or crazy, generally people wouldn't venture down this path when the correct way is still open.

"Another baseless 'intuition'?" Feng Luoyi shook his head, surprisingly not admonishing Wang Qi, just sighed: "Years ago, a similar thing seemed to happen... You avoided the path that most people deemed normal and walked a 'crooked path,' until recently when you released the Incompleteness Law and broke the notion of Completeness, I realized that the roundabout path you took years ago was actually the only correct path."

What Feng Luoyi was referring to was Wang Qi obtaining the Dao Seed Award before and using Transfinite Induction to prove the consistency of the Arithmetic System. According to what most people would think, he should have started from the "First-Order Predicate Logical System Completeness" he proved and pushed from first-order to high-order.

However, Wang Qi chose a vastly distant path—proving the absence of contradictions within the system from outside the system.

People wondered, clearly there was a clear path to Heavenly Communication, why did you insist on going outside the system and taking a detour, isn't that foolish?

At that time, only Wang Qi understood that the system couldn't achieve "without external search." The circle he took was necessary.

However, this time it's a bit different. Godel's proof idea just doesn't have the monumental significance of the Forcing Method, nor the irrefutable strength of the Forcing Method. In reality, it is still a correct path.

He could only say: "This... really is just a matter of personal preference. That thinking should not be wrong."

"Oh." Feng Luoyi nodded, unsure if he listened or not.

"By the way, teacher." Wang Qi suddenly became curious: "Who exactly proposed that idea?"

Who is the extraordinary person that thought independently of an identical approach as the great Godel?

Feng Luoyi smiled: "You should know him too. True Inheritor of Wanfa Sect, Su Junyu."

"Senior Brother Su? Does his domain include logic?" Wang Qi was astonished.

In the impression, Su Junyu's fields of study were quite diverse; it seemed like he studied almost everything, primarily focusing on probability and geometry... Hmm, thinking along those lines, it seems he wouldn't be without exposure to logic either? If one day he impulsively decided to study logic—it's very possible!

But, this truly is a fascinating coincidence. Wang Qi thought: The founding creator of the Forcing Method, Cohen, wasn't primarily a mathematics logic researcher. He was initially purely interested in the first question and wanted to dabble across fields. He personally couldn't comprehend the monumental significance of the Forcing Method.

Su Junyu is probably thinking similarly... wanting to dabble in the domain of the first question, and as a result, this step generated a major achievement.