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".

Yes, this was approximately what I meant by "encoding the word integer". Somehow it didn't occur to me that it may be difficult, or even impossible, to test real numbers for integerness. Still, I am not sure if I understand. A concrete example of a uniquely defined real number whose integerness is unknown would help.