Note that this question is part of Hilbert's program. (http://en.wikipedia.org/wiki/Hilbert%27s_program)
The first part is to formalize mathematics. Some argue that the development of ZFC, together with the work of Russell/Whitehead and Bourbaki and others has done this, or at least shown that it is possible.
The second part is to prove that the resulting axiomatization is complete. Godel's results show that this can't be done.
The third part is to show (finitistically) that formalized mathematics is consistent. Godel's results show that standard finitistic mathematics (like Peano arithmetic) can't even prove itself consistent, much less the full formalized mathematics, but of course it doesn't rule out the possibility of finitistic systems like PA+Con(ZFC). Hilbert didn't anticipate this sort of result because he didn't realize that every theory even of finitistic mathematics would be incomplete.
The fourth part is what you ask here, to show that full formalized mathematics is conservative over the finitistic part. Since ZFC proves Con(PA), but PA doesn't, Godel's results already show that this was impossible. Paris-Harrington, and Goodstein's theorems, and other examples mentioned above are nice additions only because they show that relatively straightforward statements are independent. (All three of these results are just statements that every natural number has some particular property, though the property used in Con(PA) is the property of not being the Godel code of a proof of a contradiction from the axioms of PA, which is more complicated than the properties used in the others.)
The fifth part of Hilbert's program was to give a decision procedure for all mathematical statements. This of course is far more ambitious even than the 10th problem and various other problems that have turned out to be impossible. But again, Godel's results already showed that this part was doomed, because any decision procedure could be used to generate a complete axiomatization of mathematics.