Affiliation:
1. Kharkevich Institute for Information Transmission Problems, 127051 Moscow, Russia
Abstract
A model of set theory ZFC is defined in our recent research, in which, for a given n≥3, (An) there exists a good lightface Δn1 well-ordering of the reals, but (Bn) no well-orderings of the reals (not necessarily good) exist in the previous class Δn−11. Therefore, the conjunction (An)∧(Bn) is consistent, modulo the consistency of ZFC itself. In this paper, we significantly clarify and strengthen this result. We prove the consistency of the conjunction (An)∧(Bn) for any given n≥3 on the basis of the consistency of PA2, second-order Peano arithmetic, which is a much weaker assumption than the consistency of ZFC used in the earlier result. This is a new result that may lead to further progress in studies of the projective hierarchy.
Funder
Russian Foundation for Basic Research RFBR
Subject
General Mathematics,Engineering (miscellaneous),Computer Science (miscellaneous)
Reference22 articles.
1. Kanovei, V., and Lyubetsky, V. (2023). A model in which wellorderings of the reals first appear at a given projective level, part II. Mathematics, 11.
2. Moschovakis, Y.N. (1980). Descriptive Set Theory, North-Holland. Studies in Logic and the Foundations of Mathematics.
3. Gödel, K. (1940). The Consistency of the Continuum Hypothesis, Princeton University Press. Annals of Mathematics Studies.
4. Simpson, S.G. (2009). Subsystems of Second Order Arithmetic, ASL. [2nd ed.]. Perspectives in Logic.
5. Definable sets of minimal degree;Proceedings of the International Colloquium on Mathematical Logic and Foundations of Set Theory,1970