a1 Department of Mathematics, Rutgers University, Newark, New Jersey 07102
a2 Department of Mathematics, Smith College, Northampton, Massachusetts 01063
The theory of large cardinals in the absence of the axiom of choice (AC) has been examined extensively by set theorists. A particular motivation has been the study of large cardinals and their interrelationships with the axiom of determinacy (AD). Many important and beautiful theorems have been proven in this area, especially by Woodin, who has shown how to obtain, from hypermeasurability, models for the theories “ZF + DC + ∀α < ℵ1(ℵ1 → (ℵ1) α )” and . Thus, consequences of AD whose consistency strength appeared to be beyond that of the more standard large cardinal hypotheses were shown to have suprisingly weak consistency strength.
In this paper, we continue the study of large cardinals in the absence of AC and their interrelationships with AD by examining what large cardinal structures are possible on cardinals below ℵ ω in the absence of AC. Specifically, we prove the following theorems.
Theorem 1. Con(ZFC + κ 1 < κ 2 are supercompact cardinals) ⇒ Con(ZF + DC + The club filter on ℵ1 is a normal measure + ℵ1 and ℵ2 are supercompact cardinals).
Theorem 2. Con(ZF + AD) ⇒ Con(ZF + ℵ1, ℵ2 and ℵ3 are measurable cardinals which carry normal measures + μ ω is not a measure on any of these cardinals).
(Received February 15 1985)