A proof of the Injectivity of the Continum Function

In this paper we prove that Two sets are isoplethic if and only if their power sets are isoplethic,(where isoplethic means that they have equal cardinalities) which is an old question in cardinalarithmetic and is called ICF(Injective ContinuumFunction). It is known to follow from GCH( Generalized continuum hypothesis).On this paper weprove ICF without assuming GCH or anything controversial by applying the Erdos Kaplansky Theorem which asserts that the dimension of the dual of a infinite dimensional vector space over any fieldis equal to the the cardinality of the dual.To be more specific we show that ZFC implies ICF. Finally byapplying ICF and invoking the celebrated 1964 paper of Paul Cohen, which together with his 1963paper implied the independence of CH from ZFC, we prove that Set Theory ZFC is inconsistent(if the paper of Cohen is valid本文证明，当且仅当两个集合的幂集等势时，该两集合本身等势（其中等势（isoplethic）指两集合具有相同的基数）。这是基数算术领域的一个经典问题，被称为单射连续函数猜想（Injective ContinuumFunction，简称ICF）。已知该命题可由广义连续统假设（Generalized Continuum Hypothesis，简称GCH）推导得出。本文通过应用厄多斯-卡普兰斯基定理（Erdos Kaplansky Theorem），在不假设广义连续统假设或其他存在争议性命题的前提下证明了ICF。该定理指出：任意数域上的无限维向量空间的对偶空间的维数，等于其对偶空间的基数。更具体而言，我们证明了策梅洛-弗兰克尔集合论（Zermelo-Fraenkel Set Theory，简称ZFC）可推导出ICF。最后，结合应用ICF与保罗·科恩（Paul Cohen）1964年的经典论文——该论文与其1963年的研究共同证明了连续统假设（Continuum Hypothesis，简称CH）相对于ZFC的独立性——我们证明了：若科恩的论文成立，则ZFC集合论是不一致的。