Exploring Immortality Cultivation-Chapter 901 - 230: Great Dao Sacred Sound (Part 2)
After Wang Qi appeared, Feng Luoyi and Turing both made room at the table for him, indicating that he should sit down. This meant they had already recognized Wang Qi's capability as an Arithmetician, viewing him as an equal colleague rather than a junior. Turing smiled at Wang Qi and said, "Your manifested illusion can't drink tea, so I won't pour any for you."
Feng Luoyi first asked Wang Qi, "Is everything resolved?"
"Yes, the aftermath at Gou Dabao's side is complete, and there should be a report from that side. This time, over there..."
"You don't need to tell me the specifics. I just need to know that nothing went wrong." Feng Luoyi said calmly. When Wang Qi and Gou Dabao started their empirical testing, there were already many Heavenly Swords facing that secret land. At that time, even if Gou Dabao could escape the space station-like small Heavenly Star Artifact, he would only be caught by the Heavenly Sword and completely evaporated.
Feng Luoyi briefly told Wang Qi about his recent conversation with Master Turing. Wang Qi was first amazed at what Master Turing had done and then marveled at the wonders of cosmic physical laws.
In this universe, the concept of the Von Neumann Machine appeared earlier than that of the Turing Machine. This is very fascinating. Feng Luoyi and Turing first imitated the Life Soul to standardize the Von Structure and then further mechanized it to produce a less rigorous "Turing Machine."
It is only in such a universe, where technology black boxes are particularly easy to form, that this phenomenon can be supported.
Currently, Shenzhou Cultivators understand the relationship between calculators and abacuses as follows—first, the calculator is undoubtedly an advanced version of the abacus. However, is there no transitional version between calculators and abacuses?
No one knows.
And now, Master Turing has filled this gap with the rapid progress of mathematical logic.
This research is not something that merely "fills gaps" to deceive funding. It clarifies the foundation of the calculator, allowing its development to be "down-to-earth." 𝚏𝕣𝐞𝗲𝐰𝕖𝐛𝐧𝕠𝕧𝚎𝚕.𝐜𝚘𝗺
Many theories related to calculators have gained a firmer foundation. There is also more room for research in the direction of calculators.
And according to Turing, this is just a "by-product." During his attempt to circumvent "undecidability," he made improvements to calculator theory.
But the issue Master Turing is truly contemplating is what interests Wang Qi even more.
"Are you saying you were trying to develop a method of logical reasoning to prove... prove certain problems?"
"I hoped it could bypass your incompleteness, but it seemed to fail." Master Turing spread out his manuscript, pointing, saying: "First, we define the Heavenly Element, Earth Element, and Human Element in an unknown Heavenly Transformation Formula—as we often do in the Heavenly Element Formation, using fictitious symbols to handle unknown Heavenly Transformation Formulas. Then, we assume we have two Heavenly Transformation Formulas, 'Dou' and 'Niu'—don't ask me what they specifically are; just assume there's such a thing..."
Feng Luoyi became a bit interested. This line of thought where everything is "assumed" is quite rare.
Meanwhile, Wang Qi's jaw dropped.
"This... this is obviously..."
This is the λ-calculus, another undecidable proof that once existed in Earth's history.
It is also the proof method of Church, Alan Turing's mentor, and mathematician.
Although Alan Turing eventually proved that the λ-calculus and the Turing Machine were equivalent, both were proofs of undecidability, they have completely different implications in mathematics. In subsequent developments, both Turing Machine and λ-calculus proofs opened up a grand avenue.
Thinking of this, Wang Qi looked at the slightly delicate face of Master Turing and felt a tinge of wistfulness.
Unexpected, truly unexpected. The Turing Machine and the λ-calculus proof are two sides of the same coin, diverging yet convergent. Previously, leveraging the "self-reference" of the Dongfeng, I used Turing Machine's undecidability proof, and Master Turing completed Church's line of thought.
——Great Cultivators of Shenzhou, though they bear the shadows of certain Earth figures, ultimately possess their own lives.
Church was Alan Turing's mentor. But in this realm, Master Turing is self-taught, leading the Thousand Mechanism Pavilion to suppress the myriad demons of Fusang.
——However, Turing's line of thought shouldn't match up with Church's, right? How exactly did Master Turing walk this path?
Both Carefree Practitioners are quick-witted, and their exchanges are swift. In the blink of an eye while Wang Qi was dazed, Master Turing had finished discussing his new idea. He smiled and said, "This algorithm is also quite remarkable. I tried it myself; it can prove almost all propositions and express almost all mathematical systems—it is a complete system."
This is Master Turing's contemplation on undecidability and his thoughts on incompleteness.
Feng Luoyi said, "But Wang Qi proved earlier that completeness and compatibility cannot coexist. If this is an omnipresent algorithm, then..."
"Then it must be a self-contradictory algorithm." Wang Qi traced his finger through the air, gathering colorful light, writing formulas: "For a meaningful logical system, strong expressive capability must be accompanied by unbreakable constraints. Now we substitute some obviously wrong algorithms... you see, it makes sense."
If a theory, whether right or wrong, can explain both sides and make sense no matter how you argue it, then it effectively doesn't make sense at all.
Master Turing nodded approvingly at Wang Qi's swift reaction: "You are indeed impressive. I realized this initially too. It's just that at the time, I thought I could avoid such contradictions by imposing more 'constraints.'"
"Impossible," Wang Qi shook his head and said, "Self-reference cannot be bypassed."
Self-reference is also an obstacle that the λ-calculus cannot bypass.
In this algorithm named λ, the prototype of λ is a function. When defining the λ term, it is allowed to process any function into another λ term.
Since it is "any function term," it must include the λ term itself.
So, can it process itself?
"Self-reference, what a magical concept." Master Turing praised: "I feel like the core part of our current work is all closely related to it."
The debate over the truth of "this statement is false" has existed for thousands of years, but before Wang Qi, no one recognized the terrifying implication it holds within." Feng Luoyi nodded, agreeing with Turing's view.
Incompleteness is, and so is undecidability. This "self-reference" seems to be a boundary marker in logic. It is a conspicuous boundary marker at the limits of mathematics itself.
It seems there is such a boundary, firmly limiting the development of logic.
Crossing this boundary could lead to endless loops or inherent contradictions, making it unsustainable.
How to bypass this limit was, at one time, the most concerned issue for mathematicians.
"This attempt is considered a failure," Master Turing said slowly, "The arithmetic I became known as the Wanxiang Algorithm has other meanings, which I will organize later. Next, what I will talk about is a small personal thought."
"Assumption—just an assumption," Master Turing stressed, "If we could truly bypass that limit—here, we won't discuss how specifically to bypass it; let's just say we can do it. If the Heavenly Dao became suddenly cheap, and the 'Sound of Great Dao,' 'Buddha's Chant,' and such Spirit Sense filled the world, then we could easily construct such a calculator, directly connecting to the Great Dao, providing answers to all specific problems..."
Wang Qi understood in his heart. What Master Turing described was called an "oracle machine" on Earth.
The "oracle machine," is another Turing Machine connected to "God," a black box enclosing the prophet. Suppose an oracle machine carries a prime determination oracle; besides doing everything ordinary Turing Machines can do, it can instantly determine whether a certain natural number written on a tape is a prime, without the need to calculate.
Then, could an oracle machine solve problems originally involving self-reference and contradiction?
For example, the contradiction caused by self-reference.
"This is another conjecture made recently. Can this 'self-reference' truly be circumvented?" Master Turing gave a wry smile, "Can the fictional Great Dao Sacred Sound truly resolve this issue?"
In mathematics, this method of proof often exists, where one first assumes the existence of something, then makes various logical deductions based on that premise.
The essential essence of Master Turing's deduction is—finding a hypothetical omnipotent existence, assuming it can solve problems that were previously undecidable, then seeing what issues it can actually address.
"Could the creation of our imagination truly accomplish this task?" Master Turing looked at Wang Qi and Feng Luoyi, calmly asking, "If feasible, it represents a glimmer of hope for our path. But even if our omnipotent imagination can't..."
"What are the results?" Feng Luoyi asked.
Master Turing replied with a wry smile, "Not so good."
If there exists a program Alpha for determining the halting problem (Alpha's input is a program), and we construct a new program Beta that invokes Alpha but has the exact opposite output: if Beta's input is determined by Alpha to halt, then Beta does not halt; if Beta's input is determined by Alpha not to halt, then Beta halts.
Assuming there exists an oracle machine equipped with a halting problem oracle, if the code of an ordinary Turing Machine along with its input is written on its tape, then it can instantly know whether the ordinary Turing Machine halts when encountering the indicated input, without calculation.
But even if all the "Turing Machines" in the proof are replaced with "oracle machines with number theory problem oracles," leaving everything else unchanged, the proof of incompleteness still stands!
Perhaps the limitations of human thought restricted their imagination; even the omnipotence imagined by humans could not cross this line.
Even the sound of Great Dao cannot open this door.
