Many a concrete theorem of abstract algebra admits a short and elegant proof by contradiction but with Zorn's Lemma (ZL). A few of these theorems have recently turned out to follow in a direct and elementary way from the Principle of Open Induction distinguished by Raoult. The ideal objects characteristic of any invocation of ZL are eliminated, and it is made possible to pass from classical to intuitionistic logic. If the theorem has finite input data, then a finite partial order carries the required instance of induction, which thus is constructively provable. A typical example is the well-known theorem "every nonconstant coefficient of an invertible polynomial is nilpotent".

Source : oai:arXiv.org:1308.2690

DOI : 10.2168/LMCS-9(3:20)2013

Volume: Volume 9, Issue 3

Published on: September 17, 2013

Submitted on: March 16, 2013

Keywords: Computer Science - Logic in Computer Science,Mathematics - Commutative Algebra,Mathematics - Logic

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