Enriched stratified systems for the foundations of category theory

Abstract

Four requirements are suggested for an axiomatic system S to provide the foundations of category theory: (R1) S should allow us to construct the category of all structures of a given kind (without restriction), such as the category of all groups and the category of all categories; (R2) It should also allow us to construct the category of all functors between any two given categories including the ones constructed under (R1); (R3) In addition, S should allow us to establish the existence of the usual basic mathematical structures and carry out the usual set-theoretical operations; and (R4) S should be shown to be consistent relative to currently accepted systems of set theory. This paper explains how all but parts of (R3) can be met using a system S extending NFU enriched by a stratified pairing operation; to meet more of (R3) a stronger system S∗ is introduced, but there are still some real obstacles to meeting this requirement in full. For (R4) it is sketched how both S and S∗ are shown to be consistent.

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Citations of this work

Alternative axiomatic set theories.M. Randall Holmes - 2008 - Stanford Encyclopedia of Philosophy.
Some proposals for the set-theoretic foundations of category theory.Lorenzo Malatesta - 2011 - Rivista Italiana di Filosofia Analitica Junior 2 (2):41-58.

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References found in this work

What is Cantor's Continuum Problem?Kurt Gödel - 1947 - The American Mathematical Monthly 54 (9):515--525.
What is Cantor's Continuum Problem?Kurt Gödel - 1983 - In Paul Benacerraf & Hilary Putnam (eds.), Philosophy of Mathematics: Selected Readings (2nd Edition). Cambridge University Press. pp. 470-485.
New Foundations for Mathematical Logic.W. V. Quine - 1937 - Journal of Symbolic Logic 2 (2):86-87.
Logic for mathematicians.John Barkley Rosser - 1978 - Mineola, N.Y.: Dover Publications.

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