对角线元素与不同复杂度的系统的关系

On the Relations between Diagonal Elements and Systems of Different Complexities

对角线方法是指德国学者康托尔（G．Cantor）在证明实数集不可数时所运用的一种技术方法。在这种意义上，康托尔的实数集不可数论本身便是对角线方法的一种重要运用。此外，对角线方法还有两种重要运用，即哥德尔对角线定理和汤姆逊引理的证明。

There are four kinds of relations between diagonal elements and its system.Either no diagonal element can be constructed in a system,or though diagonal elements can be constructed and can be proved to be true in a system but they are not within the system,or they can be constructed and can be proved to be true within a system,or they can be constructed within a system but cannot be proved to be true or false within the system,and there must be one of the four cases.According to the constructions of diagonal elements,real number matrixes are more complex than that of first-order logic systems,but less complex than that of real number intervals or real number,and the latters are less complex than arithmetic systems.