As for the pioneering development of set theory before Cantor, we should especially mention the Czech philosopher and mathematician Bolzano (B.),In his book Infinite Paradox, he showed that he was the first person to take positive steps towards the clear theoretical direction of establishing sets. In this book, he not only maintained the existence of real infinite sets, but also emphasized the one-to-one correspondence between what was later called two set elements. The book was not published until three years after his death, in 1851. However, in general,Before Cantor in the 19th century, the understanding and research of infinite sets remained fragmented. In the 19th century, due to the development of industrial science and natural science, the research of calculus theory and application was greatly promoted. Calculus at that time urgently required to lay its theoretical foundation, while abstract algebra at that time was actually studying groups, ringsFields and other infinite sets with special structure, and geometry has also moved to open up a new field of point set topology, so as far as the whole classical mathematics is concerned, it is urgent to establish a theoretical foundation that can encompass all mathematical branches and build on them. It is in this historical background that Cantor systematically summarized the understanding and practice of mathematics for a long time,It has created a new mathematical discipline, namely set theory. Different from the modern and modern development of set theory, the set theory founded and developed by Cantor at that time is usually called classical set theory. Because Cantor only states his theory in a simple form, neither its original concept nor its axioms are clearly listed,Therefore, it is often called simple set theory. The most important historical significance of the establishment of classical set theory is that it has realized the re expansion of mathematical research objects from finite and potential infinity to real infinity, which is the German mathematician Hausdorff,F. ) said, "It is Cantor's immortal achievement to advance from finite to infinite." The second is to provide a common theory for all branches of the whole classical mathematicsBasics.
Theoretical principles
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Although Cantor, the founder of classical set theory, only stated his theory in a simple form, neither defining the original concept nor listing its self-evident ideological provisions, as long as the content of classical set theory is summarized, Cantor's several main basic principles or thinking methods at that time are nothing more than: general principle, extension principle, one-to-one correspondence principleExtension principle, exhaustion principle and diagonal method. In addition, generalization principle and extension principle are used to create sets and determine the equality of sets, one-to-one correspondence principle and diagonal method are used to derive the concept of cardinality and determine the existence of larger cardinality, and extension principle and exhaustion principle are used to describe the generation of well ordered sets and establish the existence of real infinite research objects
