[Set Theory] 01-set theory?

"Isn't it a set? I learned from high school. Look, I don't know you when I add the word "theory! "In our impression, a set has always been the basic language of mathematics, and any branch is defined by a set. However, this situation is only decades ago,Set Theory(Set theory) was born only about one hundred years ago. You may not have imagined that set theory startsInfinityIt has also set off a new discipline"Mathematical Basics.

The Set and logic span between mathematics and philosophy, but they have generated a complex root system like other branches of the sophisticated application, making mathematics richer and more stable. As a small programmer, I am powerless and have no intention of carefully studying this huge and useless discipline. However, in view of its position in mathematics and the importance of thinking, I still seem to understand its simplest concept (not necessarily the most basic ), we can experience the spirit of modern mathematics and establish a scientific "Mathematical View ".

However, our story cannot start with the birth of the set theory, but return to back the ancient Ancient ancient Greece 2000 years ago, the age of mathematics. With the advanced prosperity of mathematics in ancient Greece, the creed of the Pythagoras school "Everything is a number" fully expresses the admiration of people. However, at that time, the number only refers to a general integer. People have always avoided the irrational number or the concept of an infinite number. FamousZhino Paradox(Zeno's paradox) not only showed people's vague understanding of infinity at the time, but also led to thinking about infinity for more than 2000 years. Archimedes has completed elementary point Computation by deliberately avoiding endless points. It has brought the exhaustion method to its extreme, but has no idea of calculus. Newton and Euler intentionally neglected infinity in the 17th and 18th centuries, which contributed to the rapid development and wide application of analytics, but the troubles of people are also increasing. The focus of the debate lies in the fact that infinity isPotential is realThe answer to this question is also necessary for the concepts of real numbers and limits. Led by WeierstrassStrict AnalysisAn infinite number is defined in the form of limits, so that people begin to face up to infinity. In modern times, people gradually realized that the concept of mathematics cannot be constructed and can be created, so infinity can also be created and defined.InternetThe (Axiomatization) storm is about to kick off.

All historical events seem to be so inevitable, but they were originally caused by accidents, and our masters appeared like this. At that time, Cantor was studying the uniqueness of trigonometric functions. The depth of the study made him realize the necessity of strictly defining irrational numbers, and thus switched to the construction of the set theory. Any proposal of new ideas will be resisted by contemporary people. klonecker of Kol also stands on the opposite side of set theory. In addition, some problems will not be proved, kol suffered from severe mental disorder and finally died in a mental hospital. However, he brought a revolution to mathematics. The phrase "the essence of mathematics lies in its freedom" is still inspiring future generations to keep moving forward.

Cancantor (1845-1918)

The storm set off by Conway has become increasingly violent, and later generations have been involved in this whirlpool. With the tide of Internet and physical chemistry, this torrents have been incorporated into the ocean of "Mathematical basics, and become the most powerful source. The open physical and chemical movement clears the vague concept of set theory and makes it a classic mathematical model. The "mathematical basis" continues to explore the logic and proof itself, and seeks the foundation of mathematics with an imposing momentum. Even many branches have gone deep into the field of philosophy. Starlight shines in the ocean of "Mathematical basics": Russell, cemerle, Feng nuiman, Godel, Coen ......, the results of "Mathematical basics" become more and more mature: metamathematics, category wheel, Force Method, and big base .......

So far, we know that set theory is far from what we know, and far from what a programmer can touch. There are several sections of descriptions in different sections, and we can't even talk about the synopsis of the story. If you are interested, you can see the thick and colorful story in any mathematical history. I reiterate that this is only the simplest concept in set theory. It is Mao, one of the "mathematical Basics". I just looked up and looked up at the existence of this peak because I could not ignore it.

 

Preface: Mathematical Logic

References:

[1] Set Theory (3rd), Thomas jech, 2006

This big blog professional book covers the vast majority of the content of set theory, and is still a standard reference book of set theory, but it is not suitable for beginners.

[2] roads to infinity: the mathematics of truth and proof, John Stillwell, 2010

It was originally a science book, but the historian has more professionalism, both interesting and rigorous, and can be used as an entry book.

[3] basic set theory, Dong Yanqi, 1988

Explains the basis of the set theory, but does not stick to the form of justice. Concise and clear language, detailed proof, reasonable content arrangement, the first choice for getting started with Set Theory in China.

[4] election principle, Zhao xishun, 2003

Starting from the history and Principles of the choice principle, we can go deep into the professional content. The first part can be used as a good Popular Science book, and the later part can be used as a professional reference book.

[5] An infinite Journey: A Cultural History of infinity (To infinity and beyond: A Cultural History of the infinite), Eli Maor, 1991

There are infinite concepts from different perspectives and disciplines, including numbers, ry, aesthetics, and the universe. Rich themes, easy-to-understand content, and a casual book.

[Set Theory] 01-set theory?