An external approach to set theory
Abstract
We begin with a primitive context with weaker logic and much simpler axioms than traditional set theory. The basic ingredients of the primitive context are essentially the object and morphism primitives of category theory. Inside this context we find `relaxed' set theory. This is considerably closer to the way sets are commonly used than is traditional axiomatic set theory, eg.~the ZermilloFraenkelChoice (ZFC) axioms, so seems to offer a better formal foundation for mathematics. This should enable more precision in work that pushes the boundaries of set theory, for example in category theory and the construction of universal objects. Next, we describe a universal wellfounded pairing, and see that it essentially satisfies the ZFC axioms. Universality implies that if a collection of settheory axioms includes the axiom of Foundation, then models appear as subobjects of this universal theory. This might, for instance, provide a way to organize the profound work on ZFC models done in the last century. It may also have consequences for the theory of large cardinals. This connects with the theory mentioned above by: an object is a `relaxed set' if and only if it is bijective to a set in the universal ZFC theory. This version provides full details, and focuses on the connection with traditional axiomatic set theory. A sequel explores consequences for category theory. These papers replace the earlier version, \emph{A construction of set theory} ( arXiv:2009.08867 )
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.01489
 Bibcode:
 2021arXiv211001489Q
 Keywords:

 Mathematics  Logic;
 Mathematics  Category Theory;
 Mathematics  History and Overview;
 03E65 (set theory) 18A05 (categories) 03B60 (logic)
 EPrint:
 36 pages. arXiv admin note: substantial text overlap with arXiv:2009.08867