Special Issue: Interactive Theorem Proving

This special issue of Mathematical Structures in Computer Science is devoted to the theme of 'Interactive theorem proving and the formalisation of mathematics'. The formalisation of mathematics started at the turn of the 20th century when mathematical logic emerged from the work of Frege and his contemporaries with the invention of the formal notation for mathematical statements called predicate calculus. This notation allowed the formulation of abstract general statements over possibly infinite domains in a uniform way, and thus went well beyond propositional calculus, which goes back to Aristotle and only allowed tautologies over unquantified statements. At the International Congress of Mathematicians in 1900, Hilbert presented a list of 23 unsolved problems, the first of which being the design of an axiomatic system for mathematics capable of internally proving its own consistency. The search for a solution to this problem had profound implications for mathematical practice since it led Cantor and others to investigate set theory and its foundations in axiomatic systems expressed in variants of predicate calculus. Some kind of consensus was reached with the first-order axiomatisation of Zermelo and Fraenkel, possibly complemented with the axiom of choice (ZFC), but mysteries remained, such as the derivability of the continuum hypothesis, or even it soundness. It was unclear whether set theory was too strong (and possibly inconsistent) or too weak (and unable to prove its own consistency). The matter was settled by Gödel, who destroyed all hopes of finding a consistent axiomatic system able to prove its own consistency. The best we could do is relative consistency proofs, such as Gödel's result that the continuum hypothesis is indeed consistent with ZFC. This is hardly satisfactory from a foundational point of view, but this line of research led to a consolidation of mathematical logic around two complementary points of view: proof theory along with the arithmetisation of syntax on the one hand and model theory within set theory on the other. Furthermore, the precise formulation of