休谟原则与弗雷格定理

Hume Principle and Frege's Theorem

弗雷格《算术的基本规律》中二阶逻辑理论FL是不一致的,在语法上可以推演出罗素悖论,在语义上,矛盾于康托尔定理,进而是不可满足的。通过仔细考察弗雷格的逻辑系统FL、FL的子系统FA以及算术还原为逻辑的推理过程,可以看出弗雷格在用公理五与概念的数的显定义推演出休谟原则后,不再实质依赖于公理五与概念的数的显定义。休谟原则与带完整二阶存在概括规则的二阶逻辑组成的系统FA是一致的,并且足以推出戴德金皮亚诺系统的五条公理,这实质上给出了不同于皮亚诺公理系统的另外一种算术公理化系统。根据自然数的定义,弗雷格实质上利用数学归纳法证明了每个自然数都有后继存在,加上后继的唯一性,弗雷格就保证了无穷多的自然数的存在。

The logical system of Frege’s The Basic Laws of Arithmetic,namely,FL is inconsistent,and able to derive Russell’s paradox.And according to Cantor’s theorem,FL can not be satisfied by any model.But the sub-systerm of FL,namely FA,which consist of full second order logic and Hume principle as its sole non-logical axiom is consistent and able to derive all Peano’s axioms.Thus FA is essentially a different axiomatization of second order arithmetic.According to Frege’s definition of natural number,he proved that every number has a successor essentially with the principle of mathematical induction,and with the theorem that every number has only one successor,the existence of infinite numbers is guaranteed.