One cannot know if a number could be written any shorter according to Gödel's incompleteness theorem

Question

I am reading Tor Nørretranders (cannot find the English version, sry) and he states that Gödel's incompleteness theorem implies that we cannot know if we can write a number any shorter (e.g. $0.3333333$... as $\frac{1}{3}$) until we find the shorter version. Until now there's no lack of clarity, but the following keeps boggling my mind: It is not possible to show that there is no shorter version to write a number.

I am having a real struggle with this statement because assumed I have got a certain number $x$ which has $y$ digits, then $y$ is a finite number. This implies I can loop through all possible statements (of course only in theory) that can be written in less than $y$ characters/digits/whatever and check if any of these instructions/algorithms outputs $x$. If there is no such algorithm within this set of algorithms which can be written using less than $y$ characters, then the number cannot be written any shorter which would be a contradiction to Gödel or at least Tor Nørretranders.

However I seriously doubt I could think of something these two geniuses couldn't.

(My approach was that I could invent an infinite number of new notations to use, such as a "new logarithm" or whatever, but somehow I just cannot think it through to the end)

PS: Sorry, couldn't think of a more appropriate title.

Answer 1

For completeness' sake I'll write down what mjqxxxx pointed out, so this question can be marked as answered (if mqjxxxx decides to write his comment as an answer I'll chose his answer as the correct one)

My fallacy was that

you can't, in general, check if a particular algorithm outputs x. You can run the algorithm, and if it outputs a wrong digit at some point, or halts early, then you know it doesn't output x. But since you can't tell if an algorithm will ever halt, you also can't tell if an algorithm won't eventually start outputting more digits of x.

(thanks mjqxxxx)