I think the author over reaches in some areas, which is what most non-mathematicians are wont to do with Gödel.
The implication that Gödel's Theorems suggest that mathematical truths exist which can never be proven at all is stretch. Godel says a particular formal system is unable, based on its axioms, to prove all true statements deriveable in the system. That doesn't mean you can't prove the statement starting with some OTHER set of axioms.
For that matter, how would it make sense to have a truth that can NEVER be proven true (in ANY axiom system)? How, exactly would we know it's "really" true?
In a more practical sense, what's the difference between a Gödel statement and an assertion we just aren't smart enough to prove? Both are unproven, and we don't know why we can't prove it (it might be false, or it might be Gödelian -- we just don't know).
My sense of the article is that is just one more example of the postmodern lunatics hijacking technical work and massaging it for their own purposes, which is to cast doubt upon scientific/mathematical epistemology in favor of ill-defined, subjective "other ways of knowing" excreta.
One last comment whilst I'm on my rant: Gödel's findings have very serious implications for meta-mathematicians, but I can't see where it poses any serious problem for ordinary mathematicians: they use theorems they prove to prove other things. Those theorems they CAN'T (or haven't yet) proven, Gödelian or not, never get used in either case, so what difference does it make in terms of the body of truth we've proven? None! What's proven is proven, and Gödel doesn't stand in the way at all.
I was also going to add the requirement, conveniently left out of the article, that the formal system must include the Peano Axioms (or their equivalent) in order that Gödel's Theorem apply, but I see you have elsewhere addressed this issue yourself. Geometry is an example of a mathematical axiom system to which Gödel does not apply.
Yes, for an incomplete system one can expand the set of axioms to satisfy completeness but one must keep expanding the set of axioms until the number of axioms is infinite in number.
This is a frequent method used to prove incompleteness of a system, by showing it requires an infinite number of axioms. Start with a finite set of axioms and then show that a new axiom must be generated and added to the set for each new proposition to be proved by the system, and that the number of such axioms can be put in one-to-one correspondence with natural counting numbers.
Not a mathematician or logician, but my understanding of it is that there are propositions such that it can be proven that there's no way to prove them either that proposition or its negation. Since either a proposition or its negation must be true, there are therefore true statements that are unprovable. That's the difference between such statements and ones for which a proof just hasn't been found yet. The former are fundamentally unprovable and the latter are not necessarily unprovable. I think that also gives us a sense in which we could possibly state that there are true statements that could NEVER be proven true. I'm not saying that Godel actually showed this, but if someone could find a statement such that both that statement and its negation were unprovable in any formal system, then we would know that there's at least one true statement that's unprovable in any formal system. We just wouldn't know whether that true statement is our original proposition or its negation.