Question Is the underlying mathematics of string theory both complete and consistent? If it is, then apparently Gödel was wrong; if it is not, then how can it be a theory of everything? Would not an endless string of metatheories be needed for sufficiency? If not, what did Gödel, Tarski, etc. miss. Dave

Accepted:
October 14, 2005

Comments

I don't know anything about string theory, but I assume that itemploys rich enough mathematics that, were we to articulate thatmathematics in a formal system, Gödel's 1931 Incompleteness Theoremwould apply to it to yield the result that, if the system isconsistent, then it is incomplete, that is, then there is somemathematical statement in the language of the system that is neitherprovable nor disprovable in that system.

You ask whether theconsistency and (hence) incompleteness of the system would conflictwith the claim that string theory is "a theory of everything". Itdepends on what "a theory of everything" means. If it means that thetheory can answer all questions about physical phenomena,then there need be no conflict: the undecidable statement of the formalsystem (the statement that can neither be proved nor disproved if thesystem is consistent) is one in the language of mathematics. It is notmaking a claim about the physical world. If, on the other hand, by "atheory of everything" one means something that can settle all questions about the physical and the mathematical worlds,then you're right that Gödel's theorem rules out such a theory: itshows that most any consistent mathematical theory will fail toanswer all mathematical questions (in particular, will fail to answervery basic mathematical questions about itself). But I suspect thatwhen people talk about string theory, it's the first sense of "theoryof everything" that they have in mind: that string theory can inprinciple settle all questions about the nature of the physical world.