Abstract
Buchholz' Omegarule is a way to give a syntactic, possibly ordinalfree proof of cut elimination for various subsystems of second order arithmetic. Our goal is to understand it from an algebraic point of view. Among many proofs of cut elimination for higher order logics, Maehara and Okada's algebraic proofs are of particular interest, since the essence of their arguments can be algebraically described as the (Dedekind)MacNeille completion together with Girard's reducibility candidates. Interestingly, it turns out that the Omegarule, formulated as a rule of logical inference, finds its algebraic foundation in the MacNeille completion.
In this paper, we consider a family of sequent calculi LIP = cup_{n >= 1} LIP_n for the parameterfree fragments of second order intuitionistic logic, that corresponds to the family ID_{<omega} = cup_{n <omega} ID_n of arithmetical theories of inductive definitions up to omega. In this setting, we observe a formal connection between the Omegarule and the MacNeille completion, that leads to a way of interpreting second order quantifiers in a first order way in Heytingvalued semantics, called the Omegainterpretation. Based on this, we give a (partly) algebraic proof of cut elimination for LIP_n, in which quantification over reducibility candidates, that are genuinely second order, is replaced by the Omegainterpretation, that is essentially first order. As a consequence, our proof is locally formalizable in IDtheories.
BibTeX  Entry
@InProceedings{terui:LIPIcs:2018:9704,
author = {Kazushige Terui},
title = {{MacNeille Completion and Buchholz' Omega Rule for ParameterFree Second Order Logics}},
booktitle = {27th EACSL Annual Conference on Computer Science Logic (CSL 2018)},
pages = {37:137:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770880},
ISSN = {18688969},
year = {2018},
volume = {119},
editor = {Dan Ghica and Achim Jung},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9704},
URN = {urn:nbn:de:0030drops97049},
doi = {10.4230/LIPIcs.CSL.2018.37},
annote = {Keywords: Algebraic cut elimination, Parameterfree second order logic, MacNeille completion, Omegarule}
}
Keywords: 

Algebraic cut elimination, Parameterfree second order logic, MacNeille completion, Omegarule 
Collection: 

27th EACSL Annual Conference on Computer Science Logic (CSL 2018) 
Issue Date: 

2018 
Date of publication: 

29.08.2018 