I am trying to use language I am not familiar with, so bear with me. If I make no sense, I try to be clearer.
Assume we are given a formal language. Assume $S$ is the set of every well-formed formula that can be made in this language.
What is the cardinality of $S$?
I would assume that it is $\aleph_0$, because every statement is a result of finitely many parts, and if every symbol is given a natural number as representation and we take a formula and convert it to a natural number by pasting the numbers for symbols together, we always get a different natural number for every formula and we always stay within the naturals.
Assume we are given a set $A$ of axioms, ie. formulas that we assume to be true.
A formula is true if there is a finite sequence of axioms and other formulas that are true that imply the formula in question. If there is no such sequence, a formula is false. A given formula is either true or false.
A proof of a formula is a demonstration that such a sequence exists or doesn't exist.
We know that given any sufficiently powerful set of axioms, there are formulas that are unprovable, ie. there is no way within the axioms to show there exists a required sequence or that there is no such sequence.
As far as I know, continuum hypothesis is an example of such a formula.
I ask, given $A$, is there a way to estimate what percentage of formulas are unprovable?
If we add an independent axiom to $A$, how does the space of all formulas change? Does it reduce the number of unprovable formulas? Does it add to the number of unprovable formulas?
Let me add one final (independent) question. Is there a way to increase the number of axioms in such a way that all formulas will become provable?