An introduction to Lorenzen's ``Algebraic and logistic investigations on free lattices'' (1951)
Résumé
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. We propose a translation (https://arxiv.org/abs/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. 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. They illustrate that lattice theory is a bridge between algebra and logic.
The preliminary manuscript contains already the main ideas and applies them to a constructive proof of consistency for elementary number theory.
Mots clés
consistency of ramified type theory without axiom of reducibility
free countably complete boolean lattice
infinitary proof theory
omega-rule
free pseudocomplemented semilattice
free semilattice
free distributive lattice
cut-elimination
consistency of elementary number theory
free countably complete pseudocomplemented semilattice
Paul Lorenzen
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