From Logicism to Metatheory

Abstract

From 1914, when Behmann first lectured on Principia in Gottingen, to 1930, when Godel proved the incompleteness of its system, Principia Mathematica played a large role in the development of modern metatheory.¹ The Principia system, with its explicit axiomatic approach to the fundamental principles of logic, was just what was needed in the early years of the 20th century to make possible the precise formulation and treatment of meta-logical questions. One might have thought, then, that at least by the time of finishing his work on Principia, Russell would have been in just the right position to appreciate such straightforward metatheoretical issues as those of the completeness and soundness of a logical system, of the independence of its axioms, and so on. But, notoriously, he seems curiously far removed from anything like modern metatheory. Russell never formulates a completeness theorem or even raises anything like a modern completeness question about his system. He even seems strangely confused about what we now take to be an entirely straightforward method of proving the independence of logical axioms. In Principles of Mathematics, Russell remarks that

[W]e require certain indemonstrable propositions, which hitherto I have not succeeded in reducing to less than ten. Some indemonstra-bles there must be; and some propositions, such as the syllogism, must be of the number, since no demonstration is possible without them.