Can uncountable be defined as unable to be generated by any algorithm I'm trying to describe the concept of uncountable to someone with even less knowledge of math than myself ;-). I came up with the idea that for any countable set, $S$, you could create an algorithm $A$, such that for any proper subset $T$ of $S$, you could generate all elements of $T$ in a finite number with a sufficient number of iterations of $A$, but that for an uncountable set, there is no such algorithm that could do this for every proper subset.
Is this a valid statement?

No.
There are uncountably many subsets of $\Bbb N$ (and all these subsets are countable). But there are only countably many algorithms. Hence you cannot describe all these sets by an algorithm.
On the other hand, there are real numbers that cannot be described by an algorithm (again, because there are more real numbers than algorithms). For such a number  $\alpha$, already the singleton set $\{\alpha\}$ is problematic for your approach.
