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A Model in Which Well-Orderings of the Reals First Appear at a Given Projective Level, Part III—The Case of Second-Order PA
- Source :
- Mathematics, Vol 11, Iss 15, p 3294 (2023)
- Publication Year :
- 2023
- Publisher :
- MDPI AG, 2023.
-
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.
Details
- Language :
- English
- ISSN :
- 22277390
- Volume :
- 11
- Issue :
- 15
- Database :
- Directory of Open Access Journals
- Journal :
- Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- edsdoj.82f6425350a642e3b3a20c5916f1ed6f
- Document Type :
- article
- Full Text :
- https://doi.org/10.3390/math11153294