A little more subtle, I think. Recall that the key idea in Godel's proof was 'Godelization" - he exhibited an algorithm which matched each sentence about arithmetic in some formal language with a particular natural number. And he also exhibited an algorithm going in the opposite direction - given any number (or formula representing a number), it would either tell you that the number does not correspond to a sentence, or it would tell you what sentence that number represents.

Now suppose we tried to duplicate this with real numbers. Our language still has only a countable number of sentences, so we still want to map sentences to integers (or naturals). So all we need is an algorithm - given any real number or formula representing a real number, determine whether that number is an integer. But there is no such algorithm. It isn't so much a difficulty in simply encoding or defining the word "integer", it is a difficulty in producing an operational or algorithmic definition of "integer".

Incidentally, returning to the original Godel and arithmetic, the only way Godel was able to produce an algorithm deciding whether a formula representing a number does or does not also represent a sentence ... the only way he could be sure his algorithm terminated, was to assume a standard model of arithmetic. I.e., count backward and you will eventually get to zero in a finite number of steps. That is, Godel's undecidable sentence could be assumed either true or false, given the axiomatization of arithmetic. But, also assuming a standard model, that sentence can only be true.