Inaccessible Set Axioms May Have Little Consistency Strength
| Venue: | Annals of Pure and Applied Logic |
| Citations: | 3 - 2 self |
BibTeX
@INPROCEEDINGS{Crosilla_inaccessibleset,
author = {L. Crosilla and M. Rathjen},
title = {Inaccessible Set Axioms May Have Little Consistency Strength},
booktitle = {Annals of Pure and Applied Logic},
year = {},
pages = {pages.}
}
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Abstract
. The paper investigates inaccessible set axioms and their consistency strength in constructive set theory. In ZFC inaccessible sets are of the form V where is a strongly inaccessible cardinal and V denotes the - th level of the von Neumann hierarchy. Inaccessible sets figure prominently in category theory as Grothendieck universes and are related to universes in type theory. The objective of this paper is to show that the consistency strength of inaccessible set axioms heavily depends on the context in which they are embedded. The context here will be the theory CZF \Gamma of constructive Zermelo Fraenkel set theory but without 2 - Induction (foundation). Let INAC be the statement that for every set there is an inaccessible set containing it. CZF \Gamma +INAC is a mathematically rich theory in which one can easily formalize Bishop style constructive mathematics and a great deal of category theory. CZF \Gamma + INAC also has a realizability interpretation in type theory whic...
