丘志宏^*

（  华为公司研究院，广东 深圳 518129；  ）

摘要： 通过递归定义自然数的方法、哥德尔的算术理论化都表明，代数系统中的元素和运算并不是分开的，而是可以相互转化。这就使得在分析代数系统时，抛开运算单独分析元素集合，可能会给分析结果带来不确定性。分析表明：1、超穷递归系统（0, +1, lim）中，去掉lim运算符将使得点等于曲线，曲线等于平面；2、递归系统（0, +1）或其他包含无穷元素的代数系统中，去掉序关系后，基数不满足三岐性；3、集合论模型和方程论模型可以相互转化，哥德尔不完备定理可以解释为方程组无解。
关键词： 集合论；一致性；完全性；悖论

Qiu Zhihong^*

（  HUAWEI Research, GuangDong ShenZhen 518129；  ）

Abstract： Both mathematical induction and G?del numbering show that, in algebraic system the elements and operations are integrate and can transform to each other. It means that analysis of a algebraic system by its elements beside its operations may lead to inconsistency or incompleteness. The analysis show that: 1, Transfinite induction system (0, +1, lim), ignore "lim" operation will lead to a point equals a curve and a curve equal a plane; 2, Induction system (0, +1), ignore the order relation, cardinality does not meet the trichotomy; 3, Set model and equations model can be transformed into each other, G?del's incompleteness theory can be interpreted as the equations has no solution.

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