2008/09/29 by Michael Pfender, Pfender, Michael · 1 citation
Computer Science · Mathematics · #03D75 #Category Theory (math.CT) #Constraint Satisfaction and Optimization #FOS: Mathematics #Logic (math.LO) #Model-Driven Software Engineering Techniques #Semantic Web and Ontologies #math.CT #math.LO #msc:03D75
paper · pdf · doi:10.48550/arxiv.0809.4970
arxiv created 2008/09/29 · openalex publication_date 2008/09/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a (minimal) Arithmetical theory with higher Order Objects, i.e. a (minimal) Cartesian closed arithmetical theory -- coming as such with the corresponding closed evaluation -- we interprete here map codes, out of [A,B] say,into these maps "themselves", coming as elements ("names") within hom-Objects BA. The interpretation (family) uses a Chain of Universal Objects Un, one for each Order stratum with respect to "higher" Order of the Objects. Combined with closed, axiomatic evaluation, this interpretation family gives code-self-evaluation. Via the usual diagonal argument, Antinomie RICHARD then can be formalised within minimal higher Order (Cartesian closed) arithmetical theory, and yields this way inconsistency for all of its extensions, in particular for set theories as ZF, of the Elementary Theory of (higher Order) Topoi with Natural Numbers Object as considered by FREYD, as well as already for the Theory of Cartesian Closed Categories with NNO considered by LAMBEK and SCOTT.