Weakly measurable cardinals

✍ Scribed by Jason A. Schanker

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 194 KB
• Volume: 57
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.201010006

✦ Synopsis

In this article, we introduce the notion of weakly measurable cardinal, a new large cardinal concept obtained by weakening the familiar concept of a measurable cardinal. Specifically, a cardinal κ is weakly measurable if for any collection A containing at most κ + many subsets of κ, there exists a nonprincipal κ-complete filter on κ measuring all sets in A. Every measurable cardinal is weakly measurable, but a weakly measurable cardinal need not be measurable. Moreover, while the GCH cannot fail first at a measurable cardinal, I will show that it can fail first at a weakly measurable cardinal. More generally, if κ is measurable, then we can make its weak measurability indestructible by the forcing Add(κ, η) for any η while forcing the GCH to hold below κ. Nevertheless, I shall prove that weakly measurable cardinals and measurable cardinals are equiconsistent.