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I'm very fascinated by the book Principia Mathematica. From what I've learned so far, Principia Mathematica 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 it's "completeness"...until Godel came along with his Incompleteness Theorem.

So my question is, if Godel's Incompleteness Theorem says that Principia Mathematica (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 Principia Mathematica?

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I don't Principia explicitly fails in something. They don't prove it's completeness, it was just that nobody thought it would be incomplete. –  MyUserIsThis Aug 5 '13 at 23:22
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You might also want to look into Wittgenstein's criticisms. Note that there were various editions of PM, some of which attempted to respond to earlier criticisms, and revise/reformulate the project, when they (Whitehead, and/or Russell) recognized oversights. –  amWhy Aug 5 '13 at 23:24
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Did it fail? Who says? It seems to me that the obsolescence of PM has more to do with the success of Zermelo's set theory as a foundational language than with anything intrinsic to PM itself. I think a better question to ask is what are the features of set theory that caused it to succeed so spectacularly, and in particular which of those features did PM lack. –  MJD Aug 6 '13 at 3:30
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Gödel used PM merely as the "best known" system of the type he wanted to discuss. But he remarks that few specific properties of PM are used in this argument, and any similar attempt to fulfill Hilbert's program would similarly fail. So I would not say "Principia failed" but instead "Hilbert's program failed". –  GEdgar Aug 6 '13 at 14:47
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2 Answers 2

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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!

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There is also a sense in which PM was a victim of its own success. PM gave a concrete demonstration of just how much mathematics could be formalized in that system, but in doing so it illustrated how much effort was required to present fully detailed, formal derivations of the results. So it succeeded in convincingly proving its point that such a formalization is possible. But, after that extremely lengthy formalization has been done one time, there is little motivation for anyone else to do a project of a similar size just to make the same point a second time. –  Carl Mummert Aug 6 '13 at 12:53
    
@CarlMummert If PM gave a concrete demonstration of just how much mathematics could get formalizaed in that system, then not a single proof not given by Russell and Whitehead could ever get written in that system. That makes no sense at all. It also doesn't make sense to speak about "approximately" how much mathematics could get written in that system either, since arithmetic has a potential (if not actual) infinity of theorems. –  Doug Spoonwood Aug 6 '13 at 22:00
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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."

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I indicate in my answer a respect in which Gödelian incompleteness is a problem for the overall project of $PM$ in a way in which it isn't for ZFC. –  Peter Smith Aug 6 '13 at 6:47
    
@PeterSmith, I guess it depends on what you mean by PM and ZFC. My answer is more from the point of view that PM and ZFC are formal systems, to be evaluated anew by each generation of mathematicians interested in using these systems. Your answer is more from the point of view that PM and ZFC were (historically) mathematical and philosophical projects, situated in a particular intellectual context. So, I think both our answers are valid, but they just emphasize different points of view. Of course, you sort of point this out with your point number (2), but just in different words. –  goblin Aug 7 '13 at 3:40
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