Is the set of all information infinite or finite?

If it is finite, then can it be used to completely define anything infinite? At some point as we pursue finite information ever further, wouldn’t we fall of the edge of the world—so to speak—unable to speak any longer about the infinity we are endeavoring to describe.

If it is infinite, like the set of all true, or well-formed formulas in Principia Mathematica (PM) is, then can’t information be coded in such a way as to answer that question? If so, what do you think is the likelihood that a self-referential statement could be correctly coded in information/code similar to the set of PM Gödel Numbers proving that information/code can generate a proof that itself is false?

In other words, does not Gödel’s Incompleteness Theorem apply to any set of infinite information/code in such a way that information/code cannot prove itself in any but trivial applications.