When the continuum has cofinality ω₁

Arnold William Miller and Karel Libor Prikry

Vol. 115 (1984), No. 2, 399–407

DOI: 10.2140/pjm.1984.115.399

Abstract

In this paper we consider models of set theory in which the continuum has cofinality ω₁. We show that it is consistent with ¬CH that for any complete boolean algebra B of cardinality less than or equal to c (continuum) there exists an ω₁-generated ideal J in P(ω) (power set of ω) such that B is isomorphic to P(ω) mod J. We also show that the existence of generalized Luzin sets for every ω₁-saturated ideal in the Borel sets does not imply Martin’s axiom.

Mathematical Subject Classification 2000

Primary: 03E35

Secondary: 03C62, 03E50, 06E05, 54H05

Milestones

Received: 17 March 1983

Revised: 6 June 1983

Published: 1 December 1984