Two geniuses
Battle
In the late 19th and early 20th centuries, mathematics was in a period of unprecedented prosperity.
However, the development of things has two sides, and the more prosperous it seems on the surface, the easier it is to hide the crisis behind it - the third mathematical crisis.
Set theory, one of the most important foundations of mathematics, has penetrated into many branches of mathematics, but some people have found fatal paradoxes in set theory. It directly impacts the disciplines of mathematics and logic, which are known for their rigor, and shakes the standards of credibility of traditional mathematical concepts, mathematical propositions, and mathematical methods.
It's like suddenly one day we're told that the multiplication tricks we've memorized since childhood may all be wrong.
One of the most representative is Russell's "haircut paradox" proposed in 1919.
A village barber announced the principle that he shaved the faces of all those who did not shave themselves, and only the faces of such people in the village. Paradoxical nature: "Does the barber shave his own face?" "If he doesn't shave his face, then he should shave his face according to principle; if he shaves his face, then he doesn't conform to his principles."
In order to defend the rigor and scientific nature of mathematics, a giant stepped forward - Hilbert.
David Hilbert was a famous German mathematician
Hilbert has been called "the last all-rounder in mathematics". His best "non-constructive proofs" and "law of exclusion" are among the most distinctive and controversial methods of proof in mathematics.
Take a chestnut:
"Non-structural proof": There are 100 seats in a classroom, but only 99 students. It can be concluded that there must be a vacancy, but we cannot be sure where that vacancy is located. The "law of exclusion" is better understood: one thing is either true or false.
Inspired by "non-constructive proofs" and "exclusionary laws", Hilbert proposed the famous "Proof Theory Plan", also known as the "Hilbert Plan", in 1920.
Project Hilbert is somewhat similar to the C language used by programmers to code, in order to formalize all mathematics—all mathematical representations should be in a mathematical language with uniform standards and used according to a set of strict rules.
Moreover, no matter how esoteric and complex the mathematical conjectures, as long as we follow this method, it is only a matter of time before the "truth comes out".
Supermodel Jun said so much, what is the significance of Hilbert's plan?
First, after the abandonment of natural language, mathematical expressions became more rigorous. For example, "There is a set that is empty" →
。 Second, mathematics is complete. In other words, only if this mathematical statement is correct, we must be able to prove its authenticity in this way. Third, mathematics is consistent. That is, there are no self-contradictory mathematical statements, which ensures that the results we obtain without violating logic make sense. There will not be a single statement, it is both true and false.
Hilbert's ultimate idea was to find a reasonable algorithm that would be used to determine the authenticity of all mathematical statements.
This sounds like a good plan, but in fact, Hilbert is "lifting a stone and dropping it on his own feet."
On August 8, 1900, Hilbert gave a lecture at the Second International Congress of Mathematicians known as the "Highest Point of Mathematics in the 20th Century", in which Hilbert asked 23 mathematical questions.
While mathematicians all over the world were working on solving these "23 mathematical problems", Gödel was still thinking about the Hilbert plan, and the most surprising thing was that he actually overturned the "Hilbert plan".
Kurt Gödel was a mathematician, logician and philosopher
It can be said that Gödel began to like to tinker with mathematical logic because of Hilbert.
After reading Hilbert's Principles of Mathematical Logic, he simply chose a topic from his dissertation as the subject of his doctoral dissertation— "Are true propositions all provable in formal systems?" ”
The conclusion of Gödel's doctoral dissertation is the little-known "Gödel's theorem of completeness" – the completeness of the first-order predicate calculus. But this theorem has a fatal drawback: first-order logic is too limited, and it cannot even define natural numbers, let alone do arithmetic.
After Gödel discovered this problem, he spent a year doing deeper research, but came to a completely opposite conclusion - Gödel's incompleteness theorem. (which contains the first and second theorems)
First theorem: Any formal system of first-order logic and elementary number theory has a proposition in which it can neither be proved true nor as negative.
That is to say, if we can do arithmetic in a mathematical system, then either the system is self-contradictory, or there are some conclusions that are true and we cannot prove them.
The second theorem , if the system S contains elementary number theory , its non-contradiction cannot be proved in S when it has no contradiction.
To put it in layman's terms, Gödel produced a logical contradiction in Hilbert's method of proving mathematical statements, obtaining a paradox of truth and falsehood, and negating the consistency proposed by Hilbert.
So why does Hilbert have a feeling of "lifting a stone and dropping it on his own feet"?
For Gödel's process of overturning Hilbert's plan and proving these two theorems is what Hilbert's plan refers to: formalization. And Gödel also perfectly solved the second of Hilbert's "23 mathematical problems".
First, Gödel followed Hilbert's line of thought by expressing all mathematical statements in strict symbolic terms; then replacing them all with natural numbers (Gödel's digitization); and finally by the recursiveness of mathematical induction; however, Gödel found that the result he obtained was still a natural number.
To put it bluntly, this mathematical statement states itself—itself—itself.
The supermodel jun gave a very simple example to the model friends, "This sentence is wrong", so is it right or wrong?
Similarly, the "haircut paradox" that caused the third mathematical crisis was a self-referential process of a logical explosion, which also represented Gödel's successful overthrow of Hilbert's plan.
Mathematical expression of the "barber paradox": If there is a set A = {x | x ∉ A }, then does A ∈A hold? If it is true, then A ∈ A and does not satisfy the characteristic properties of A; if it does not, A satisfies the characteristic properties.
I have to admit that genius is genius. Gödel cleverly used the "self-referential paradox" to construct a proposition that itself could not be proved, leading to these conclusions:
If it is true, then we have a true and unprovable proposition, and the system is not complete. If it is false, then there is a proof of it, so that it should be true, indicating that the system is contradictory and inconsistent. Finally, we assume that the system is consistent, but this would contradict the "unprovability of propositions" mentioned above.
This is the idea of Gödel's first theorem of incompleteness, the first two points (the first theorem) are aimed at solving the completeness of the Hilbert plan, and the third point (the second theorem) is the solution to consistency.
Gödel's god-like operation not only draws people from the "23 mathematical problems", but also completely smashes the "Hilbert Plan", and if we assume that mathematics does not contradict itself, we must admit that mathematics is incomplete, that is to say, that there are mathematical propositions that are indeterminate: we can neither prove them true nor false.
But there were still many mathematicians at the time who believed that although Hilbert's dream was broken, this did not threaten the normal development of mathematics.
Because Hilbert left an eternal truth to his descendants at the moment of his retirement.
We need to know, we will know.
We must know, we must know.