Where, specifically, did Principia Mathematica fail? I'm very fascinated by the book Principia Mathematica. From what I've learned so far, PM set out to be, essentially, the bible of mathematics and logic, from which all mathematical and logical ideas can be derived. As I understand it, the book was initially a success in that no one challenged its "completeness"… until Gödel came along with his incompleteness theorem.
So my question is, if Gödel's incompleteness theorem says that PM (and any other such work) can never fully encompass all of math and logic, what specific ideas/theorems/proofs/etc. exist in mathematics and logic that can't be derived from PM?
 A: By Goedel's second incompleteness theorem, the system set out in Principia cannot prove its own consistency.
Edit. By the way, I've always thought rather poorly of the argument that "Principia failed because of the incompleteness theorems." Firstly, this argument can be applied to any foundational system. "ZFC failed because of the incompleteness theorems." Except that it didn't, because everyone (well, almost everyone) loves ZFC.
I think that, in reality, foundational systems are chosen not only for their level of completeness, but also for their usability. From what I've heard, the Principia system was unwieldy and inconvenient to use; so, if we're going to take the position that Principia Mathematica failed, then we should probably take the position that: "it failed because it was inconvenient."
A: Perhaps we should distinguish two questions:

(1) Does $PM$ fail as a philosophical/foundational project: if so, why?
(2) Does $PM$ in some sense fail as a mathematical project: if so, why?

Re (1): It is usually said that the philosophical project of PM was to defend a version of logicism, by showing that all arithmetical truths, all truths of classical analysis, and more, can be deduced from purely logical assumptions plus definitions (where the definitions cash out notions like "is a natural number" in purely logical terms). In other words, Russell and Whitehead are trying to complete Frege's logicist project (but this time, in a consistent logical framework!).
$PM$ fails, read as attempting this philosophical project. For a start, its Axiom of Infinity and Axiom of Reducibility are very difficult to defend as logical axioms. And then -- a fatal blow -- Gödel's first incompleteness theorem shows that the ambition of capture even all arithmetical truths in a single (decidably) axiomatized system must fail. [Aside: ZFC isn't propounded with the same all-encompassing logicist ambitions as $PM$, so isn't challenged in this way by Gödelian incompleteness.]
Re (2): The ramified type theory of PM is mathematically messy to work with in various ways. Not just inconvenient, but messy in a way that suggests that it is not implementing its leading idea in the mathematically best way. And indeed as Ramsey pointed out long ago, the mathematical project of $PM$ doesn't need some of the complications (the complications which require the Axiom of Reducibility and which seem to be there for more philosophical reasons -- i.e. to give a one-size-fits-all treatment of a whole range of paradoxes, logical and semantic). So let's separate out one key mathematical idea, and we get a near descendant of $PM$, preserving Russell's key idea, namely (Simple) Type Theory -- which is alive and well and not a failure at all!
