I know that the Cantor snake diagonalization argument is used to show that a set is countable. As an extension, I'm wondering whether the same argument can be used to prove that a set is uncountable? Why can't we weave a snake diagonally around the real numbers to show that the real numbers are countable?
Does the fact that the real numbers are not well-defined mean that the real numbers are not countable? Is a set uncountable iff it is not well defined?
There are two important points that must be made here.
Firstly, the set of real numbers is well-defined. I am worried because you think that it is somehow not well-defined. Think about it: If $\mathbb{R}$ was not a well-defined set, how could one prove theorems about the reals? $\mathbb{R}$ can be defined as the completion of the set of rational numbers wrt. convergence of Cauchy sequences. See e.g. https://en.wikipedia.org/wiki/Construction_of_the_real_numbers for more about the construction of the reals.
Secondly, the "snake" argument – more commonly called dovetailing – is a constructive argument used to establish the existence of a bijection between a set $S$ and $\mathbb{N}$. The "snake" is a representation of the bijection. One can only show that a set $S$ is uncountable by showing that there can be no bijection between $S$ and $\mathbb{N}$.
Your question indicates several confusions. I will try to straighten them out.
The real numbers are well defined, so your claim that they are not can't be the basis for any conclusion.
There are several reasonable ways to define the real numbers, starting with the positive integers (assuming you accept that they are well defined). The easiest to understand is probably the definition of a real number as an infinite decimal.
Then Cantor's diagonal argument proves that the real numbers are uncountable.
I think that by "Cantor's snake diagonalization argument" you mean the one that proves the rational numbers are countable essentially by going back and forth on the diagonals through the integer lattice points in the first quadrant of the plane. That argument really proves that the countable union of countable sets is countable.
But there is no way to "weave" through the real numbers that will end up expressing them as a countable union of countable sets - since Cantor proved they are uncountable.
