Comment by RugnirViking
9 days ago
im not very good at all this, having just a basic engineers education in maths. But the sentence
> There are a countably infinite descriptions, as at the end every description is text
seems to hide some nuance I can't follow here. Can't a textual description be infinitely long? contain a numerical amount of operations/characters? or am I just tripping over the real/whole numbers distinction
> Can't a textual description be infinitely long?
That's a good question. The usual answer is no.
The idea is that every book/description that has ever been written can be seen as string of finite length over a finite alphabet. For example, the PDF of the book is a file, i.e., a string of finite length over the byte-alphabet.
Another way to think about it, that every book/description has to have been written. Writing started some time in the past. Since then a finite amount of time has passed. Assuming one writes one character per second at most, one obtains an upper length on the number of characters in the book. This implies that it is finite in length.
That being said, one way to define the real numbers is to start with infinite sequences of rational numbers. Next, one defines when they converge against the same number. A real number x is then defined as the classes of sequences that converges against x. The set of infinite sequences of rational numbers is uncountable infinite. That's where the cardinality of the real numbers at the end of the day comes from.
The reason I bring this up is because one can view an "infinite sequence of rational numbers" as "infinitely long textual description". So your question really scratches the core of the problem.