arXiv 0803.17152008-03math.GM

连续统假设的反证及用集合逼近确定连续统的基数

Disproof of the Continuum Hypothesis and Determination of the Cardinality of Continuum by Approximations of Sets

Slavko Rede

备注：: This version is an extension of the previous one. In the sections 6 to 12 we build the tool for determining the cardinality of the power set of natural numbers inside the set theory developed in the previous version. The sections 1 to 5 remain unchanged with the exception of a few stylistic corrections

摘要：

集合论是在集合的近似基础上发展起来的，记作AS。在作为所有集的集合存在，但罗素和康托的悖论的论点失败。分离公理、替换公理和基础公理是无效的。ZF的所有其他公理都是有效的，并且所有的基本集合，如补、交和笛卡儿积都存在，尽管补与ZF中的集合不完全相同。所有集合的集合可以配备有近似拓扑（Ta）。每个集合在Ta中是闭的，每个函数在Ta中是连续的。这意味着连续统假设是错误的。包含在Ta中完美的子集的集合具有最大的基数。一个简单的观察表明，良序的概念必须以与ZF略有不同的方式定义。我们证明了一个集合可以是良序的当且仅当它没有完美子集。因此，任意集合的基数总是可比较的，而不需要假定选择公理。跟随最小无限基数w的基数是2w，3w，...，w^2，2w^2，3w^2，>.，w^3.每个都比前一个具有更大的基数，这在ZF中不是这种情况。紧接着这些基数后面的不是w^w，w ^w不是良序集，而是某个良序基数k，这个基数后面是基数2k，3k，.，k^2，2k^2，3k^2，.，k^3....，最大基数是P（w），并且不是良序集。良序集的共尾性是2或w。唯一的正则基数是0，1，2，w和P（w）。所有其他枢机主教都是单数。唯一的强极限基数是w。唯一的不可达基数是P（w）。强不可达基数的枢机主教是不存在的。

英文摘要：

A set theory is developed based on the approximations of sets and denoted by AS. In AS the set of all sets exists but the argument for Russell's and Cantor's paradox fail. The Axioms of Separation, Replacement and Foundation are not valid. All the other axioms of ZF are valid and all the basic sets, such as complement, intersection and cartesian product, exist although complement is not quite the same set as in ZF. The set of all sets can be equipped with the topology of approximations (Ta). Every set is closed and every function is continuous in Ta. This implies that the Continuum Hypothesis is false. The sets containing a subset which is perfect in Ta are of the greatest cardinality. A simple observation shows that the concept of well-ordering must be defined in a slightly different way than in ZF. We prove that a set can be well-ordered if and only if it has no perfect subset. Therefore the cardinalities of arbitrary sets are always comparable without assuming the Axiom of Choice. The cardinals following the smallest infinite cardinal w are 2w, 3w, ..., w^2, 2w^2, 3w^2, >..., w^3.... each being of greater cardinality than the previous one, which is not the case in ZF. Immediately after these cardinals does not follow w^w which is not a well-orderable set but some well-ordered cardinal k, and this one is followed by the cardinals 2k, 3k, ..., k^2, 2k^2, 3k^2, ..., k^3...., etc. The greatest cardinal is P(w) and is not a well-orderable set. The cofinality of a well-ordered set is either 2 or w. The only regular cardinals are 0, 1, 2, w and P(w). All other cardinals are singular. The only strong limit cardinal is w. The only inaccessible cardinal is P(w). Strongly inaccessible cardinals do not exist.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。