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Mathematics > Logic

arXiv:1804.02439 (math)
[Submitted on 6 Apr 2018]

Title:Dathematics: A Meta-isomorphic Version of 'Standard' Mathematics based on Proper Classes

Authors:Danny A. J. Gomez-Ramirez
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Abstract:We show that the (typical) quantitative considerations about proper (as too big) and small classes are just tangential facts regarding the consistency of Zermelo-Fraenkel Set Theory with Choice. Effectively, we will construct a first-order logic theory D-ZFC (Dual theory of ZFC) strictly based on (a particular sub-collection of) proper classes with a corresponding special membership relation, such that ZFC and D-ZFC are meta-isomorphic frameworks (together with a more general dualization theorem). More specifically, for any standard formal definition, axiom and theorem that can be described and deduced in ZFC, there exists a corresponding `dual' version in D-ZFC and vice versa. Finally, we prove the meta-fact that (classic) mathematics (i.e. theories grounded on ZFC) and dathematics (i.e. dual theories grounded on D-ZFC) are meta-isomorphic. This shows that proper classes are as suitable (primitive notions) as sets for building a foundational framework for mathematics.
Subjects: Logic (math.LO)
MSC classes: 03B10, 03E99
Cite as: arXiv:1804.02439 [math.LO]
  (or arXiv:1804.02439v1 [math.LO] for this version)
  https://doi.org/10.48550/arXiv.1804.02439

Submission history

From: Danny Arlen de Jesús Gómez-Ramírez [view email]
[v1] Fri, 6 Apr 2018 19:51:49 UTC (15 KB)
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