Most references to
Cantor's diagonal argument, such as Gorgonzola's, seem to give the appearance that it is impossible to create a
list of
real numbers that contains its own modified diagonal. To clarify (or confuse) matters, I will show that such a list is possible but that list will still be
incomplete.
Example:
f(0)=.10000000000...
f(1)=.19999999999...
f(2)=.12931234134...
f(3)=.77799999999...
f(4)=.14159231415...
f(5)=.31245931234...
(Boldface digits for clarification)
By increasing the diagonal digits by one (wrapping around from 9 to 0), we get the number 0.200000000... which is the same as 0.19999999... = f(1).
Now, every number in this list, except the first, has the digit 9 in at one place or another. Therefore the list doesn't include numbers such as 1/9 = 0.11111... or 1/3 = 0.333333... . It's still incomplete.
Using the digit 9 all the way in the diagonal in unavoidable, as the only real numbers with dual representation are those with infinite trailing nines / zeroes.