Communications of the ACM
Formally Verified Mathematics
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From the point of view of the foundations of mathematics, one of the most significant advances in mathematical logic around the turn of the 20th century was the realization that ordinary mathematical arguments can be represented in formal axiomatic systems in such a way their correctness can be verified mechanically, at least in principle. Gottlob Frege presented such a formal system in the first volume of his Grundgesetze der Arithmetik, published in 1893, though in 1903 Bertrand Russell showed the system to be inconsistent. Subsequent foundational systems include the ramified type theory of Russell and Alfred North Whitehead's Principia Mathematica, published in three volumes from 1910 to 1913; Ernst Zermelo's axiomatic set theory of 1908, later extended by Abraham Fraenkel; and Alonzo Church's simple type theory of 1940. When Kurt Gödel presented his celebrated incompleteness theorems in 1931, he began with the following assessment:
"The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so one can prove any theorem using nothing but a few mechanical rules. The most comprehensive formal systems that have been set up hitherto are the system of Principia Mathematica on the one hand and the Zermelo-Fraenkel axiom system of set theory (further developed by J. von Neumann) on the other. These two systems are so comprehensive that in them all methods of proof used today in mathematics are formalized, that is, reduced to a few axioms and rules of inference. One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case..."4
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