Title: Algebraic and logistic investigations on free lattices,
Abstract: Lorenzen's "Algebraische und logistische Untersuchungen über freie
Verbände" 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 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 lattice 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.