An Introduction to Lorenzen's 'Algebraic and Logistic Investigations on Free Lattices' (1951)

Artikel i vetenskaplig tidskrift, 2025

Lorenzen's article has immediately been recognised as a landmark in the history of infinitary proof theory. We propose a translation and this introduction in order to publicise its approach and method of proof, without any ordinal assignment. It is best known for providing a constructive proof of consistency for ramified type theory without axiom of reducibility 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. It proposes a definition of semilattices, of distributive lattices, of pseudocomplemented semilattices, and of countably complete boolean algebras as deductive calculi, and constructs conservatively the respective free objects over a given preordered set. It illustrates that lattice theory is a bridge between algebra and logic: the construction of an element corresponds to a step in a proof. The fruitfulness of its use of the omega-rule is immediately recognised by Schutte. It triggers the analysis by Ackermann (1951) of the infinitary inductive definition of the accessibility predicate with the goal of proving transfinite induction up to ordinal terms beyond & varepsilon;(0).