Predicative Categories of Classes Michael Alton Warren Summary: Predicative Categories of Classes Michael Alton Warren October 28, 2004 Contents Abstract iii Acknowledgments iv Introduction v 1 -Pretopoi and Constructive Set Theories 1 1.1 -Pretopoi and Dependent Type Theory . . . . . . . . . . . . 1 1.1.1 -Pretopoi . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Dependent Type Theory . . . . . . . . . . . . . . . . . 2 1.1.3 Soundness and Completeness . . . . . . . . . . . . . . 4 1.2 Constructive Set Theories . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Notation and Axioms . . . . . . . . . . . . . . . . . . 6 1.2.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . 8 1.2.3 The Category of Sets . . . . . . . . . . . . . . . . . . . 12 1.2.4 Axioms of Infinity . . . . . . . . . . . . . . . . . . . . 14 2 Predicative Categories of Classes 16 2.1 Categories with Class Structure . . . . . . . . . . . . . . . . . 16 2.1.1 Axioms for Categories with Basic Class Structure . . . 16 Collections: Mathematics