数学原始概念的新选择

New Choice of the Primitive Concept of Mathematics

选择元素作为数学的原始概念.元素满足基本、统一、独立、确定、分明的5条公设.元素可构成序列和集合.集合满足5条生成新公理:空起查后继、后继能加列、终结才确集、后继可永存、理想设穷竭.给出了无限集合的新定义:元素在现实中永有后继而且在理想中一个不漏的集合.用无限序列来反映无穷的进程,用无限集合来反映无穷的终结,强调两者不能混淆.还给出了5条自然数序集的生成新公理:下界0存在,正向1次序,素量n标准、后继w无限,上界∞存在.与皮亚诺公理相比,反映了数的度量性和无穷设终结.将数学哲学的研究和数学基础的建构紧密结合,力求3组公设公理的逻辑关系清楚,形式简明优美,语言通俗易懂.

In this paper, element is selected as primitive concept of mathematics. The element needs to satisfy 5 postulates defined, so that element is a basic, unified, independent, definite and clearly demarcated object, on which to base sequence and set from elements. Moreover, 5 axioms of set＇s constructing are defined： The empty is the initial point of the set and its successor is checked; The successor can be arranged in the array. The termination of arranging can just end the set. The successor can exist forever in reality. The infinite successor can be ended in ideal. Meanwhile, new definition of infinite set is defined： The set, the arranging of its elements without one being leaked in the ideal and have successors forever in reality. The infinite sequence is used to reflect the infinite＇ s process and the infinite set is used to reflect the infinite＇s termination. The two above-mentioned are not obscure. 5 axioms of the natural number sequence-set formulation are defined newly： The 0 is the lower bound; The 1 is the basic order; The n is standard quantity; The w is infinite successor; The ∞ is the upper bound. The measurement of number and the termination of infinite should be reflected in our axiom. Studying philosophy of mathematics and constructing foundation of mathematics are connected closely, it is hoped to make a try to 3 groups of postulates and axioms be clear logic relations and concise graceful forms.