Read on formally undecidable propositions of Principia Mathematica and Related Systems

Read on formally undecidable propositions of Principia Mathematica and Related Systems

 

A formula is a finite sequence of natural numbers, and a proof is a finite sequence of finite sequences of natural numbers. formulas are sequences of natural numbers-> expressible in pmclass-sign: A formula of PM with exactly one free variable of type natural numbersk = {n belongs in | not provable (RN (N ))} K is the set of numbers where the formla RN (n) that you get when you insert N into its own formula RN is improvable. for a specic natural number Q. we will now prove that the theorem RQ (q) isundecidable within PM. we can understand this by simply plugging in the denitions: RQ (Q), S (Q), Q 2 K,: provable (RQ (q), in other words, RQ (q) States/I am improvable. "assuming the theorem RQ (q) were provable, then it wowould also be true, I. e. because of (1): provable (RQ (q) wocould be true in contradiction to the assumption. if on the otherhand: RQ (q) were provable, then we wowould have Q 62 k, I. e. provable (RQ (q )). thatmeans that both RQ (q) And: RQ (q) wocould be provable, which again is impossible. P is the system that you get by building the logic of PM on top of peano axioms.

I still don't fully understand, especially in the preface, why is the conflict? However, I still understand the previous sections. I read this article to translate the original German version into an English version, and only the first two parts are translated, but the last two parts are not translated. The translator pointed out that directly reading the original article has three difficulties:

1. The original file is German.
2. Godel uses many symbols of that era.
3. Godel assumes that his readers have sufficient mathematical knowledge when writing this article.

To overcome these three difficulties, the translator uses the following methods to solve these problems:

1. English Translation
2. Use Symbols that people are familiar with to replace symbols that people are not familiar.
3. For some places that people may not be familiar with, the author uses hyperlinks to point to these areas.

The background of this article is probably like this: in the early 20th century, people were surprised to find that since there were several systems of justice (Forms), it seems that all the theorems are a combination of those principles. At that time, there were two primary systems of justice: Principia Mathematica (PM), and zermelo-fraenkelian.

Because all the theorems are a combination of several theorems, people naturally wonder if all the theorems in this system can be expressed by these theorems, or, how can we prove it with these principles? Just as people were excited and amazed at the perfection of mathematics, Godel published this article, in which he proved that in all forms and systems, at least one proposition is not verifiable, several examples are provided. His proof method is very interesting. First, we need to make the proof system belong to PM. He used natural numbers instead of all symbols to do this. He does not directly prove that there is an unverifiable proposition in all formal systems. Instead, he first proves that the formal system under two conditions has an unverifiable proposition, and then further discusses it, after the two conditions are removed, it proves that there are unverifiable propositions in all forms and systems.

The above English section is my notes for reading this article. It is a very important concept. It is useful to write down these concepts during this article.

Other gains from reading this article:

1. For articles that require intensive reading, you must first read them roughly and then start reading them again. Then you will be aware of what is easy to understand and where is time-consuming. In this way, you will not be too worried when you encounter problems during intensive reading.
2. Reading classic articles takes more time and allocates more time. If you just browsed it in a hurry, the gains would not be great.
3. Reading these articles will not have immediate results. It is a long-term accumulation.
4. If you read this article under the guidance of experts, it will be much better. If you discuss it, the effect will be much better.
5. I am hesitant to send an email to anyone who has translated this article. The email is correct. Maybe one day, I will send such an email.