论二阶算术的理论解释

On Theoretic Interpretations of Second-Order Arithmetic

直观而言,自然数有无穷多个,每一个都可从初始元(设为0)出发,通过有限步后移而达到(有限可达性),自然数全体构成一个良序集。从结构的角度刻画自然数集所遇到的最大困难是如何既表征自然数集的无穷性又表征每一个自然数的有限可达性。

The essay studies the theoretie interpretations of Dedekind-Peano’s axioms of (second-order) arithmetic. In my view, the philosophical foundation of a theoretic interpretation of arithmetic is to provide an answer to the question what nature numbers are and DP axioms should be the adequacy condition for such answers. The deadly bolw to logicism would be this condition rather than logical antinomies such as Russell’s paradox. Frege’s thesis that numbers are objects places type-reducing functions into the core of the theoretic interpretations of arithmetic, Russell’s paradox show that Frege can only appeal to the second-order theories of type-reducing functions, and that Fregean Arithmetic is an adequate theoretic interpretation of the second-order arithmetic relies on a fortunate fact that the type-reducing function satisfying Hume Principle has a atandard semantic model.