In chapter 8 of Shoenfield's matheamtical logic[1967], He proves that The formula of P which states that P is consistent is not a theorem of P, where P stands for Peano Arithmetic. And then He says, without a proof, that this result can be extended to more general theories. I don't really know what 'more general theories' exactly mean.
'more general theories' mean recursive consistent extensions of P?
Thank you.
Certainly it applies to any (consistent, computably axiomatizable) extension of $\mathsf{PA}$ (that's the standard modern notation for first-order Peano arithmetic). It also holds for many vastly weaker systems of arithmetic, as well as theories which are not about arithmetic (directly, anyways) at all, like $\mathsf{ZFC}$.
On the other hand, unlike the first incompleteness theorem (whose extent is discussed in section $4$ of this paper of Beklemishev), there are reasonably-strong systems to which it doesn't apply. These have been studied by Willard; briefly speaking, they are systems which do not prove that multiplication is always defined, which kills the argument of the second incompleteness theorem due to a "size blowup" issue. See this answer of mine for more about this.
The short version, though, is: the second incompleteness theorem applies to any "natural" theory which can "do basic arithmetic."
OK, but what does that actually mean? (The following is unfortunately pretty technical, but I've tried to keep it still readable.)
Here's one fairly general - indeed, the most general I'm aware of - way to tackle the second incompleteness theorem:
Suppose $T$ is any (consistent, computably axiomatizable) theory which interprets Robinson arithmetic via some formula tuple $\Phi$. Then "$\Phi$'s version of" the consistency statement of $T$ is not $T$-provable.
This takes some unpacking. First of all, Robinson arithmetic is just a very weak theory of arithmetic; if you like, you can replace that with $\mathsf{PA}$ for simplicity for now. The real issue is around interpretations. The idea here is initially to handle things like $\mathsf{ZFC}$, which are not directly about the natural numbers but within which we can implement the natural numbers. The definition is fairly technical, but basically an interpretation of one theory $\mathsf{A}$ into another theory $\mathsf{B}$ is a tuple $\Psi$ of formulas in the language of $\mathsf{A}$ such that $\mathsf{A}$ proves "The structure described by $\Psi$ satisfies $\mathsf{B}$." For example:
The usual construction of the rational numbers as equivalence classes of ordered pairs of integers with nonzero second coordinate - with $(a,b)$ intuitively standing for $a\over b$ - amounts to an interpretation of the theory of $(\mathbb{Q};+,\cdot)$ in the theory of $(\mathbb{Z};+,\cdot)$.
Writing down the formulas defining the finite ordinals and ordinal addition and multiplication yields an interpretation of $\mathsf{PA}$ into $\mathsf{ZFC}$. (Note that actually $\mathsf{ZFC}$ proves a lot more than $\mathsf{PA}$ for this interpretation, but it doesn't matter that we've "overshot" $\mathsf{PA}$; extra strength is fine.)
So far so good - "interprets $\mathsf{Q}$" is also the standard strength condition for the first incompleteness theorem. But now we come to the new bit, which is basically how we express "$T$ is consistent" in the language of $T$. We already know how to do this in the context of arithmetic, but what if our theory in question is not about arithmetic at all, at least on the face of things? For example, $\mathsf{ZFC}$ falls into this category. Or, maybe we're looking at some complicated theory of geometry (much stronger than simple Euclidean geometry of course) which can in fact talk about arithmetic in some roundabout way; how should consistency be expressed there?
The idea is to use the interpretation we already have as a kind of "translation" mechanism. If we let $Con(T)$ be the usual formulation of "$T$ is consistent" in the language of arithmetic, the interpretation $\Phi$ whips up a corresponding sentence $\mathfrak{Con}(T)$ in the language of $T$ itself such that we can run Godel's argument "through" $T$'s version of arithmetic and "back again" into $T$, with the result that $T$ does not prove $\mathfrak{Con}(T)$.
Perhaps more cleanly, the statement of the second incompleteness theorem above can also be phrased as:
Suppose $T$ is any consistent, computably axiomatizable theory. Let $Con(T)$ be the usual statement in arithmetic asserting the consistency of $T$. Then $T$ does not interpret $\mathsf{Q}+Con(T)$.
