The Canonicity of Homer

Canonicity!

We describe an abstract proof-theoretic framework based on normal-form proofs, defined using well-founded orderings on proof objects. This leads to robust notions of canonical presentation and redundancy. Fairness of deductive mechanisms – in this general framework – leads to completeness or saturation. The method has so far been applied to the equational, Horn-clause, and deduction-modulo cases.

The Homer System

Ground Canonicity

They are not capable to ground a canonicity of universal consistency. Abstract We explore how different proof orderings induce different notions of saturation. We relate completion, paramodulation, saturation, redundancy elimination, and rewrite system reduction to proof orderings.

Non-canonicity of Mv-algebras

The varietyMV of all MV-algebras is shown to be non-canonical in a strong sense. Specifically it is shown that the canonical extension of the Chang algebra, K2, is not an MV-algebra. As a consequence, no non-finitely generated variety of MV-algebras is canonical.

The Canonicity of Point-free Topology

Properties of topological spaces are not absolute, as they may depend on the ambient mathematical universe. For instance, the answers to some questions in classical point-set topology hinge on non-standard set-theoretic axioms, while in intuitionistic mathematics even such basic properties as the Heine-Borel compactness of the closed interval depend on the ambient constructive variety. We show ...