>>14210876LS theorem is more complicated than you might think, but let's actually go and comprehend what it implies for the reals: The LS theorem says you have a countable model, which means that all elements that you can write with symbols are in some countable set, and that is the important part, all elements you can write with symbols.
>And so, are there reals that can't be described by your normal trig, hyperbolic trig, exp, log, integrals, etc.; functions?Indeed, there are, and by LS they are the majority of reals. These should come as no surprise, since we also know that the majority of reals are trascendental irrationals and yet they are quite hard for us to find.
Now, by "written down" I meant in a first-order language. An example of an indescribable number, under this definition, would be inspired by Rayo's number: Define Rayo(n) as the least natural that can't be described with n symbols in first-order language, then let
Then x is an indescribable number since if it were describable, it would be in at least N symbols, but then their (N+100)th 1 would be in Rayo(N + 100) position, so you could describe Rayo(N+100) in less than N + 100 symbols, contradiction.
