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第一章包含我們會敍述過的公理法,而且它也包含著最直接地從公理得出的一系列定理,讀者有必要注意到掌握這些定理的證明的會部重要性,檢查甚至記憶希爾伯脫的系統的全部公理,是頗不困難的,可是如果不學會實際上運用這些公理,就是在這些公理的基礎上嚴密邏輯地來證明這些定理,那麼對於數學的發展是毫無用處的。希爾伯脫的敍述每次重版都是向著更容易理解與更完備這一面改變的,雖然,在其中到現在還有不少的在證明上的空白,讓讀者來補充它們,究其實,這樣的情況有力地減低了該書的教育價值,問題不僅在於若干省略了的證明是相當困難的地方,還更重要是別的,初學者甚至在已經作出證明之後,未必能夠有把握地了解他的證明在邏輯的方面是否無可責難,或者在其中某些地方難入從直觀所假借來的假設:所以編者和譯者抱定目標,用附在該書後面(原書403-488頁)的附錄來補足敍述的空白。
The first chapter contains axioms that we will describe, and it also contains a series of theorems most directly derived from axioms. It is necessary for the reader to pay attention to the significance of the clubs that master the proofs of these theorems, and to check and even remember them. The whole axioms of Bert’s system are not difficult, but if these axioms are not actually applied, it is on the basis of these axioms to prove these theorems in a logical manner, which is of no use for the development of mathematics. Hiberthor’s narrative changes every time the revision is more easily understood and more complete. Although, there are still many gaps in the proofs, and let the readers supplement them. Actually, in this way, The situation has effectively reduced the educational value of the book. The problem lies not only in the fact that a number of omitted proofs are quite difficult, but also more importantly in others. Beginners may not be able to understand him with certainty even after proof has been made. To prove that there is no censure in terms of logic, or in some places it is difficult to assume the hypothesis borrowed from intuition: so the editor and the translator hold the target, and use the appendix attached to the book (pages 403-488). To make up for the narrative gap.