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Pré-Publication, Document De Travail Année : 2017

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.
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Dates et versions

hal-01629913 , version 1 (07-11-2017)
hal-01629913 , version 2 (20-09-2023)

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  • HAL Id : hal-01629913 , version 1

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Thierry Coquand, Stefan Neuwirth. An introduction to Lorenzen's ``Algebraic and logistic investigations on free lattices'' (1951). 2017. ⟨hal-01629913v1⟩
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