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
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Received on Sun Dec 16 2007 - 05:33:34 PST