An introduction to Lorenzen's "Algebraic and logistic investigations on free lattices" (1951)

This article proposes an historical and mathematical introduction to Lorenzen's "Algebraische und logistische Untersuchungen \"uber freie Verb\"ande". These "Investigations" appeared in 1951 in The journal of symbolic logic. They have immediately been recognised as a landmark in the history of infinitary proof theory, but their approach and method of proof have not been incorporated into the corpus of proof theory. More precisely, the admissibility of cut is proved by double induction, on the cut formula and on the complexity of the derivations, without using any ordinal assignment, contrary to the presentation of cut elimination in most standard texts on proof theory. We propose a translation (arXiv:1710.08138) and this introduction with the intent of giving a new impetus to their reception. We also propose a translation of a preliminary manuscript, "A preorder-theoretic proof of consistency", with the kind permission of Lorenzen's daughter, Jutta Reinhardt. The "Investigations" are best known for providing a constructive proof of consistency for ramified type theory without axiom of reducibility. Lorenzen does so by showing that it is a part of a trivially consistent "inductive calculus" that describes our knowledge of arithmetic without detour. The proof resorts only to the inductive definition of formulas and theorems. He proposes furthermore a definition of a semilattice, of a distributive lattice, of a pseudocomplemented semilattice, and of a countably complete boolean lattice as deductive calculuses, and shows how to present them for constructing the respective free object over a given preordered set. This work illustrates that lattice theory is a bridge between algebra and logic. The preliminary manuscript, given as an appendix, contains already the main ideas and applies them to a constructive proof of consistency for elementary number theory.