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

TL;DR

Lorenzen's 1951 work introduced innovative proof-theoretic methods, including cut admissibility without ordinal assignments, and provided constructive consistency proofs for ramified type theory, bridging algebra and logic through lattice constructions.

Contribution

The paper presents a new approach to proof theory, notably a cut elimination method without ordinal assignments, and constructs free lattices as deductive calculuses, advancing the understanding of infinitary proof systems.

Findings

Proved cut admissibility via double induction without ordinal assignments.

Constructed free lattices as deductive calculuses for various lattice types.

Provided a constructive consistency proof for ramified type theory.

Abstract

Lorenzen's ``Algebraische und logistische Untersuchungen \"uber freie Verb\"ande'' appeared in 1951 in The Journal of Symbolic Logic. These ``Investigations'' have immediately been recognised as a landmark in the history of infinitary proof theory. Their approach and method of proof have not been incorporated into the corpus of proof theory. Lorenzen proves the admissibility of cut 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 this introduction with the intent of giving a new impetus to their reception. The ``Investigations'' are best known for providing a constructive proof of consistency for ramified type theory without axiom of reducibility. They do 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. They propose furthermore a definition of a semilattice, of a distributive lattice, of a pseudocomplemented semilattice, and of a countably complete boolean algebra as deductive calculuses, and show how to present them for constructing conservatively the respective free object over a given preordered set. They illustrate that lattice theory is a bridge between algebra and logic for which the construction of an element corresponds to a step in a proof. We shall describe the history of their reception, which focusses mainly on the omega-rule. The fruitfulness of this device is immediately recognised by Kurt Sch\"utte. 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 epsilon_0, which is also taken over by Sch\"utte (1952).