A plausible account of necessity is that p is necessary provided that p can be proved in the correct logical system K and p is possible provided that its negation cannot be proved. Assuming K is axiomatizable and proves enough of the axioms of arithmetic, this account can be shown to be incorrect.
Fix any sentence s in K. It follows from Goedel's Second Incompleteness theorem that there is no K-proof of s's being K-unprovable (for if there were such a proof, then it would follow that there is a K-proof of K's consistency, since if K is inconsistent, then every sentence, including s, can be proved in K). But on the account of modality under consideration, this means that it is possible that s is K-provable, i.e., it is possible that s is necessary.
In other words, this account of modality implies that every sentence is possibly necessary. But it is absurd to think that 0=1 is possibly necessary!
I think much the same reasoning can be used to disprove Swinburne's account of necessity, since where we are not dealing with directly referential rigid designators, Swinburne's account agrees with the provability account.
I am skirting over distinctions between s and its Goedel number, but I think that's a mere technicality to work out in greater precision.
This makes for a nice way to see a relationship between the two incompleteness theorems. The first one tells us that not everything true is provable. From the second we learn that not even everything necessary is provable.