Von Neumann's first paper, co-authored with Fichte, is a generalization of the Fehn's theorem of Chebyshev polynomial root method, and the date is 1922. At that time, Von Neumann was less than 18 years old. Another article discusses uniformly dense series written in Hungarian. The choice of topic and the simplicity of proof technology reveal the intuitive combination of von Neumann's algebraic skills and set theory.
1923, when von Neumann was a university student in Zurich, he published a paper exceeding the ordinal number. The first sentence of the article bluntly stated that "the purpose of this article is to make Cantor's ordinal number concept concrete and precise". His definition of ordinal number has been widely adopted.
It is von Neumann's wish to explore axiomatization vigorously. From about l925 to l929, most of his articles tried to carry out this axiomatic spirit, even in theoretical physics research. At that time, his expression of set theory was particularly informal. At the beginning of his doctoral thesis on axiomatic system of set theory in 1925, he said, "The purpose of this thesis is to give an axiomatic exposition of set theory logically and irreproachable."
Interestingly, von Neumann foresaw the limitations of any form of axiomatic system in his paper, which vaguely reminded people of the incompleteness theorem proved by Godel later. Professor frankl, a famous logician and one of the founders of axiomatic set theory, once commented on this article: "I can't insist that I have understood everything, but I can safely say that this is an outstanding work, and I can see a giant through him".
1928, von Neumann published the article "Axiomatization of Set Theory", which is an axiomatic treatment of the above set theory. The system is very concise. It uses the first type object and the second type object to represent the set and the properties of the set in naive set theory. It takes a little more than one page to write the axioms of the system, which is enough to establish all the contents of naive set theory, thus establishing the whole modern mathematics.
Von Neumann's system may give the first foundation of set theory, and the finite axiom used has a logical structure as simple as elementary geometry. Starting from axioms, Von Neumann's ability to skillfully use algebraic methods to deduce many important concepts in set theory is simply amazing, which has prepared conditions for his interest in computers and "mechanization" proof in the future.
In the late 1920s, von Neumann participated in Hilbert's meta-mathematics project and published several papers to prove that some arithmetic axioms were not contradictory. 1927 The article "On Hilbert's Proof" has attracted the most attention, and its theme is to discuss how to get rid of contradictions in mathematics. The article emphasizes that the question put forward and developed by Hilbert and others is very complicated and has not been answered satisfactorily at that time. It is pointed out that Ackerman's proof of eliminating contradictions is impossible in classical analysis. Therefore, von Neumann gave a strict finiteness proof of the subsystem. It seems that this is not far from the final answer Hilbert wants. Just then, 1930 Godel proved the incompleteness theorem. Theorem assertion: In an incongruous formal system containing elementary arithmetic (or set theory), the incongruity of the system is unprovable in the system. At this point, von Neumann can only stop this research.