I studied Cantor's Diagonal Argument in school years ago and it's always bothered me (as I'm sure it does many others). In my head I have two counter-arguments to Cantor's Diagonal Argument. I'm not a mathy person, so obviously, these must have explanations that I have not yet grasped.
My first issue is that Cantor's Diagonal Argument (as wonderfully explained by Arturo Magidin) can be viewed in a slightly different light, which appears to unveil a flaw in the argument. If we assume that s_f
is an image of f
at some index n
, then it does not make sense to define s_f
as $s_f=(s_1,s_2,s_3,…,s_n,…)$ where $\begin{equation} s_k = \begin{cases}0 & \mathrm{if\ } f(n)_n = 1\\1 & \mathrm{if\ } f(n)_n = 0\end{cases}\end{equation}$
since then the $n$th element of $s_f$ would be defined as the opposite of itself. Since Cantor's Diagonal Argument uses this definition that wouldn't make sense if s_f
has an index, then s_f
must not have an index, and from there it seems obvious that they would reach the conclusion that s_f
is not an image of f
. Isn't that begging the question?
My second issue is more complicated, and less articulate, but basically that when I attempt to put numbers into Cantor's Diagonal Argument, I could demonstrate that the "missing" element was the within a constant distance from the last element in the "series", which means all of the infinite other numbers must be before it, which means no matter how long you count, you'd never reach it. For example, if one puts these in the most obvious order of "counting" 0000..., 1000..., 0100...., 1100..., 0010... then the element to be found is obviously the element where all $s_k = 1$, which would be the "last" element in that ordering. But that also seems to apply to the counting numbers, which also seems to violate Cantor's Arguments.