1999/11/28 by Sakaé Fuchino, Fuchino, Sakaé, Stefan Geschke +5
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis #math.LO
paper · pdf · doi:10.48550/arxiv.math/9911230
published as Ann. Pure Appl. Logic 110 No. 1-3 (2001) 89--105
arxiv created 1999/11/28 · openalex publication_date 1999/11/28 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The following results are proved: (a) In a model obtained by adding aleph2 Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebras without the weak Freese-Nation property consistent with GCH. (c) Under some consequences of the negation of 0^#, the weak Freese-Nation property of (P(omega),subseteq) is equivalent to the weak Freese-Nation property of any of C(kappa) or R(kappa) for uncountable kappa. (d) Modulo consistency of (alephomega+1,alephomega)-->(aleph1,aleph0), it is consistent with GCH that the assertion in (c) does not hold and also that adding alephomega Cohen reals destroys the weak Freese-Nation property of (P(omega),subseteq)