Exploring Immortality Cultivation-Chapter 369 - 122 Paradox

In a certain sect, there was an alchemist who once boasted, "My alchemical skills are excellent and renowned throughout Shenzhou. I shall provide alchemical services for all within the Immortal Dao who cannot concoct their potions. I will only serve them!" Those who came to him for alchemy were naturally those who couldn't concoct their own potions. However, one day, feeling that his cultivation had stagnated, he instinctively opened his pill furnace. Then, could he concoct a potion for himself?

This is the famous Alchemist's Paradox. When reduced to a low-version without magic, it's still the Alchemist's Paradox. If one prefers a more sophisticated expression, it is also equivalent to, "Russell's Paradox."

Define property P(x) as "x does not belong to x." Now suppose that a class A is determined by property P—meaning "A={x|x?x}." The question then arises: Does A belong to A? First, if A belongs to A, then A is an element of A; therefore, A has property P, and by property P, A does not belong to A. Secondly, if A does not belong to A, meaning A has property P and since A consists of all classes with property P, A belongs to A.

This is the insurmountable issue logicists could not circumvent: Is the whole also a part of itself?

This very question led to the inconsistency within Set Theory.

There were methods to skirt around this paradox, and Wang Qi knew two: the Zermelo-Frankel Axiomatic System and the von Neumann-Bernays-Godel System. The former restricts the methods of operating sets, whereas the latter separates sets and classes. These are efforts—or struggles—made by Earth's mathematicians on this issue.

"As long as Set Theory remains inconsistent, the Mathematics Lord will not admit defeat—Okay, I admit, even if Set Theory were consistent, it would still be meaningless to him. To him, it's just a hollow concept, a word game." Wang Qi curled up in front of his study, reading the Mathematics Lord's paper again: "And to prove Set Theory's completeness… am I foolish? Knowing this path leads to darkness because it is inherently incomplete, yet trying to prove it?"

Although this comprehensive paper by the Mathematics Lord was more of a provocation, a declaration, the author was still one of the greatest mathematicians in Shenzhou, and it had academic value.

For Wang Qi, the greatest value of the paper was that it organized the most critical problems currently encountered in Shenzhou's mathematics logic.

He held a pen in his hand but didn't write; instead, he twirled it with his fingers.

A small amount of logic could almost deduce all of mathematics—but in the eyes of the Mathematics Lord, that was simply not mathematics. In such a case, whatever Wang Qi achieved would be seen by the adversary as "an even more ludicrous trick."

On Earth, such academic disputes often quickly devolve into endless nitpicking and personal attacks. Mathematics cannot throw an irrefutable result through experiments, and disputes eventually degrade into slander. For instance, Earth's historians of mathematics quite rudely criticized David Hilbert and Ruiz Brouwer for lacking the character to "keep disputes within academic boundaries." This is the result of consuming too much ginseng and chicken—it's ineffective and excessive nourishment.

"In Shenzhou, people probably resolve such disputes in a much more violent manner?" For some reason, Wang Qi felt somewhat excited at the thought of a grand battle between the Wanfa Sect and the Carefree factions.

The essence of this academic dispute was actually to boost one's supporters' confidence and then seek out and exploit weaknesses in the opponent's other theories, striking their academic standing, or doing similar research to prove complete superiority in mathematics over the opponent, until one side's confidence was utterly depleted and they succumbed to surrender, allowing the other side victory.

Of course, Earth's academic circles also had a somewhat disreputable method—waiting for the older generation to die out, allowing time to "prove everything." However, in Shenzhou, this tactic was rather impractical.

"If simply attacking the opponent's other theories to reduce their academic standing... Well, it's not undoable, but that's just instigating hatred."

Wang Qi quickly realized the huge advantage he held as a transmigrant.

The mathematics he had studied was at least half a century ahead of Shenzhou's. He didn't even need to fully digest the opponent's theories; simply comparing their theories with his own knowledge and pinpointing the differences in details could easily accomplish the task of "finding faults."

However, such destructive efforts rarely earned academic respect and were apt to attract animosity. Wang Qi still counted on the Mathematics Lord and Teacher Feng to lead, while he gradually built up his resources and bided his time.

As for completely overwhelming the opposition... forget it. The Mathematics Lord was one of the strongest mathematicians in history, arguably number one. Such a complete domination was not only difficult, as it was tantamount to independently deriving the entirety of the mathematics system he had learned. Wang Qi conservatively estimated that by the time he completed this task, he would have achieved transcendence himself.

After pondering it through, Wang Qi sadly realized that essentially, what he needed to do hadn't changed—perhaps, he just needed to speed things up a bit.

The gap between the two sides was too significant.

"However, it's precisely because of this that it's interesting!" Wang Qi stood up to stretch his body and then pushed open the door, heading towards the library at the Divine Capital Station.

In his eyes, in addition to the intent to battle, there was an even greater pure radiance.

He had studied mathematics far beyond the existing systems of Shenzhou, so he knew that whether it was the Mathematics Lord or the Master of Calculation, whether it was the Li Sect or the connecting Sects, they would all be utterly defeated in this debate, with no victors.

Godel's Incompleteness Theorem would indeed shatter the Master of Calculation's perfect and optimistic dream, but in the future, the foundation of mathematics would still be rebuilt by logic.

Therefore, in this debate, the temporary wins and losses were not the focus.

When the greatest Arithmeticians in the world compete on the same stage, what brilliance could he himself achieve?

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Unbeknownst to Wang Qi, a new storm was brewing.

Deep within the Wanfa Sect, the Mathematics Lord threw the monograph he was holding onto the ground and laughed, "So many years have passed, yet it's still these convoluted things—Mr. Xi, are you perhaps planning to revert to the Ancient Method over these years?"

Even with his excellent Qi Cultivation Skills, Xi Baiche's complexion darkened with anger. Pang Jialai was not just verbally mocking them; just moments ago, he had forcibly driven Xi Baiche's disciples out of the Ten Thousand Immortals Illusion Realm with his formidable mathematics skills.

This was an explicit declaration of war.

"Pang Senior still has his old charm—just like your crude papers."

"Heh." Pang Jialai laughed, "I'm here actually for one thing."

Xi Baiche, unsure of the other's intentions, looked at him with suspicion.

"I won't be staying here long this time. But I can't allow you to desecrate the Mathematics Dao." A glint of sharpness flashed in Pang Jialai's eyes, "So, how about we make a bet?"

"What bet?"

"If I lose, I'll go back. If you lose, you'll go for me."

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Mr. Hilbert and Mr. Brouwer… this truly is a tragedy. Mr. Brouwer, known for his fiery temper and thoughtless remarks, was the first to use the Ginseng Rooster. Unfortunately, after inadvertently insulting the entire German mathematics community, the usually kind Mr. Hilbert finally lost his temper and united all the scholars working in Göttingen, except for Einstein, to criticize the other side. He thought Mr. Brouwer would quickly reignite, allowing him to comfortably withdraw the Divine Skills. But Mr. Brouwer was excessively indignant—he fell ill from anger and even accused Mr. Hilbert of losing his mind… and then… and then…

This truly is a tragedy.