Any two noticeable R numbers a and b can be an open interval (a,b) of infinitely many R numbers, that cannot be separated form each other by any representation, and each R number can be represented only by aleph0 different representations. Is it right ? If what i wrote holds, than please tell me how can Cantor use some aleph0 representation to prove that there are more numbers then this aleph0 representation? What i clime is: Cantor cannot start by using the assumption saying that he produces its new diagonal number (which is not in the list) from a complete list of R numbers, because a complete list of R numbers has no representation of any kind, therefore no information of any kind, and therefore there is no basis for any conclusion like: R numbers are not countable | R numbers are countable . An example: 0 . ? ? ? ? ? ... 0 . ? ? ? ? ? ... 0 . ? ? ? ? ? ... 0 . ? ? ? ? ? ... 0 . ? ? ? ? ? ... We cannot use the assetion that we have a complete list of R numbers, because in this case we cannot find any number in the diagonal, therefore no list, no information, no result (or proof if you like). Please show by Math language, how can we come to some result by using no information as a meaningful input?