Does the Number of Elements in Natural Number Set Equal to That in Even Number Set?——New Scheme for Infinite Theory (1)

This paper briefly introduces conflicting debates and puzzles of the infinite in mathematic,logical and philosophical circles in the 20th century.It proposes an original plan in the infinite study: the ideal and the reality must be distinguished as studying the existence of infinite;the process and the end must be distinguished as studying the structure of infinite;to approach a limit and to measure must be distinguished as studying the quantity of elements in infinite set and fundamental metric and transformed metric must be distinguished as studying the metric.And it provides metric principle and methods referring to infinite.This paper also analyzes correspondence and studies Cantor's definition of the infinite set:The set one-to-one corresponds with its proper subset and his conclusion:The quantity of elements in natural number set equals to it in even number set.This paper still points out Cantor's mistake in infinite area,in which the process and the end,the set and the sequence,and the injection and bijection are confused.Finally,it figures out that the conclusion: Parts equal to the unity in infinite area,which violates the Contradiction Law,must be discarded.