康托尔实数的局限性

康托尔为我们建立了集合论，并且证明了实数的不可数性，但是其中留下了很多疑点． 1．—个实数能在每—个集合论模型中出现的充分必要条件是它是可以被集合论来定义的．那些在集合论模型中不出现的实数，我们可以把他们叫做看不见的实数． 2．在实数的十进位无穷小数表示法中有些是我们能确切地知道它的第几位是什么，但是对另外的一些实数我们对它们就只能有模糊的认识，也就是说它的第几位是什么我们不可能全部知道．我们可以把他们叫做写不出的实数． 3．由于Cantor关于实数是不可数的证明不是构造性的证明，而是用所谓的归谬证法．它们中有很多是看不见写不出的实数．因此说它们是虚拟的实数． 4．虚拟实数就像银行中的虚拟货币，你可用它来买东西，它可从—个户头转拨到另—个户头，但是钱的实体是不存在的。这个现象也让我们对某些数学工具的合法性挺出质疑．我们用对角线法来证明实数的基数比自然数的基数大。但是我们并没有真正有效的地构造出那么多的实数．因此我们没有办法来确切地定义它们．也可以说它们中的绝大多数是不可以定义的．在一般的情况下虚拟实数是不可以个别地使用的．

The limitations of Cantor's real numbers

Cantor proved that the number of all real numbers is uncountable but from the LST theorem of model theory a countable theory should have a countable model. Set theory is a countable theory. It should have a countable model. Real munber is a set. It should be countable in the model. This is so call Skolem paradox. In this paper we further explain this phenomena. Our conclusions are.
1. A necessary and sufficienr condition for a real munber to appear in. every set theory model is that it is definable by a set theory formula. Therefore some of the real numbers can be absent from a set theory model.
2. In the expressions of infilfite decimal of real numbers some of them we can know their digits one by one but for the others we can only have fussy, knowledge. It is to say we can not know all of their digits.
3. Since the tmcountablity proof of all real numbers given by Cantor is not comutructible, the major part of the real nmnbers are fictitious. It is like the fictitiotm money in the bank. We can use it to do shopping. It goes from one account to another but the money body never exists. We use diagonallization method to prove that the cardinal number of real numbers is greater than the cardinal nmnber of natural numbers, but we do not really produce such amount of real numbers. So now we have trouble to identify （define） them. As a result most of them are indefinable. Because of the undefinability in general fictitious real numbers can not be used individually.