Suppose we have a formal system, F, in which the second incompleteness theorem can be applied. If we were able to prove, using some higher level system, F’, that F is consistent, do we not have the issue of determining whether F’ is consistent? If F’ is inconsistent, then the result that F is consistent is meaningless. But then if we have an F’’ which proves the consistency of F’, we have to determine if F’’ is consistent, and so this process is never ending. So can we ever know if something like PA is consistent?
My second question is about incompleteness and independence. Assuming a formal system, F, is consistent, how can we know that a given statement is independent? As I understand it, the first incompleteness theorem only shows the existence of a statement which is unprovable and true, not that there is a statement in which it being true or false does not affect the consistency of the system. Must all sufficiently powerful consistent systems have independent statements or can they all just be variations of the Gödel sentence?
(1) "As I understand it, the first incompleteness theorem only shows the existence of a statement which is unprovable and true, not that there is a statement in which it being true or false does not affect the consistency of the system." No. The first incompleteness theorem applied to an appropriate consistent theory $T$ will show that there are pairs of sentences $G$ and $\neg G$ such that $T$ can't prove or disprove either of them. One will be true and one false. Either of them can be added to $T$ preserving consistency.
(2) "If we were able to prove, using some higher level system, $F'$, that $F$ is consistent, do we not have the issue of determining whether $F$ is consistent?" But note it can well be the case that, while $F'$ can prove something that $F$ can't, we are antecedently more confident of the consistency of $F'$ than $F$. For example, a proof has been announced of the consistency of Quine's deviant set theory NF. The proof -- assuming it works -- can be regimented in standard ZFC. Why is the proof worth having? Because we are antecedently more (a lot more?) confident that ZFC is consistent than that NF is.
(3) On the consistency of PA, you think it is inconsistent? ;)
It reflects our intuitive conception of the natural numbers; it can be proved consistent by various other theories you should have no doubts about (weak set theories, say); the few sane attempts to show it is inconsistent (Nelson's for example) don't work. What more could you want?
The interest of the second theorem applied to PA is this: if PA can't prove the consistency of PA, it can't prove the consistency of a theory stronger than PA. And putting it crudely, that sabotages "Hilbert's Program", the hopeful idea that we can use finitary reasoning of the kind we could encode in PA to prove at least the consistency of stronger theories, like ZFC or NF say.
You'll find elaborations of these points in any standard text -- take a look at e.g. the short version of mine, the freely downloadable Gödel Without (Too Many) Tears.