# Re: Cantor's Diagonal

 Hi Folks, I joined this list a while ago but I haven't really kept up.  Anyway, I saw the reference to Cantor's Diagonal and thought perhaps someone could help me. Consider the set of positive integers: {1,2,3,...}, but rather than write them in this standard notation we'll use what I'll call 'prime notation'.  Here, the number m may be written as m = n1,n2,n3,n4,... where ni is the number of times the i'th prime number is a factor of m. Thus: 1 = 0,0,0,0,0,... 2 = 1,0,0,0,0,... 3 = 0,1,0,0,0,... 4 = 2,0,0,0,0,... 5 = 0,0,1,0,0,... ... 28 = 2,0,0,1,0,0,0,... ... If we now apply the diagonal method to this ordered set, we get the number: D = 1,1,1,1,1,... Has this just shown that the set of positive integers is not denumerable? I can see that one may complain that D is clearly infinite and therefore should not be in the set, but consider the following... Let's take the original set and reorder it by exchanging the places of the i'th prime number with that of the number in the i'th position.  (i.e. First switch the number 2 with the number 1 to move it to the first position. Then switch 3 with the number -- now 1 -- in the 2nd position. Then 5 with the 1 which is now in the 3rd position. Etc...) Because we are just trading the positions of the numbers, all the same numbers will be in the set afterwards. The set is now: 2  = 1,0,0,0,0,... 3  = 0,1,0,0,0,... 5  = 0,0,1,0,0,... 7  = 0,0,0,1,0,... 11= 0,0,0,0,1,... ... Now instead of adding 1 to each 'digit' of the diagonal, subtract 1.  This will ensure that the diagonal number is different from each of the numbers in the set. Thus, D = 0,0,0,0,... But this is the number 1 which we know was in the set to begin with.  What happened to it? I would suggest that the diagonal method does not find a number which is different from all the members of a set, but rather finds a number which is infinitely far out in the ordered set. If anyone can find where I've gone wrong, please let me know. Dan Grubbs --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---