Algebraic and logistic investigations on free lattices

TL;DR

This paper revisits Lorenzen's 1951 work on free lattices, highlighting its innovative proof methods in infinitary proof theory and its constructive approach to consistency in ramified type theory.

Contribution

It provides a translation and analysis of Lorenzen's original work, emphasizing its novel proof techniques and constructive methods in proof theory and lattice theory.

Findings

Double induction proof of cut admissibility without ordinal assignment

Constructive proof of consistency for ramified type theory

Definition and construction of free lattices as deductive calculuses

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, but their approach and method of proof have not been incorporated into the corpus of proof theory. More precisely, 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. This translation has 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 formulae 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 the respective free object over a given preordered set. This translation is published with the kind permission of Lorenzen's daughter, Jutta Reinhardt.