TY - GEN
T1 - Large model constructions for second-order ZF in dependent type theory
AU - Kirst, Dominik
AU - Smolka, Gert
N1 - Publisher Copyright:
© 2018 Copyright held by the owner/author(s). Publication rights licensed to the Association for Computing Machinery.
PY - 2018/1/8
Y1 - 2018/1/8
N2 - We study various models of classical second-order set theories in the dependent type theory of Coq. Without logical assumptions, Aczel's sets-as-trees interpretation yields an intensional model of second-order ZF with functional replacement. Building on work of Werner and Barras, we discuss the need for quotient axioms in order to obtain extensional models with relational replacement and to construct large sets. Specifically, we show that the consistency strength of Coq extended by excluded middle and a description operator on well-founded trees allows for constructing models with exactly n Grothendieck universes for every natural number n. By a previous categoricity result based on Zermelo's embedding theorem, it follows that those models are unique up to isomorphism. Moreover, we show that the smallest universe contains exactly the hereditarily finite sets and give a concise independence proof of the foundation axiom based on permutation models.
AB - We study various models of classical second-order set theories in the dependent type theory of Coq. Without logical assumptions, Aczel's sets-as-trees interpretation yields an intensional model of second-order ZF with functional replacement. Building on work of Werner and Barras, we discuss the need for quotient axioms in order to obtain extensional models with relational replacement and to construct large sets. Specifically, we show that the consistency strength of Coq extended by excluded middle and a description operator on well-founded trees allows for constructing models with exactly n Grothendieck universes for every natural number n. By a previous categoricity result based on Zermelo's embedding theorem, it follows that those models are unique up to isomorphism. Moreover, we show that the smallest universe contains exactly the hereditarily finite sets and give a concise independence proof of the foundation axiom based on permutation models.
KW - Consistency strength
KW - Coq
KW - Dependent type theory
KW - Second-order set theory
KW - Sets-as-trees model constructions
UR - https://www.scopus.com/pages/publications/85044316635
U2 - 10.1145/3167095
DO - 10.1145/3167095
M3 - Conference contribution
AN - SCOPUS:85044316635
T3 - CPP 2018 - Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with POPL 2018
SP - 228
EP - 239
BT - CPP 2018 - Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with POPL 2018
A2 - Felty, Amy
A2 - Andronick, June
PB - Association for Computing Machinery, Inc
T2 - 7th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2018
Y2 - 8 January 2018 through 9 January 2018
ER -