Talk:Measurable cardinal
From Wikipedia, the free encyclopedia
I love your comment on the mistake in previus version. Another way is to use a model in which κ is not measurable but it becomes measurable after a forcing. Kunen has such a model. 157.181.80.93 11:15, 11 June 2006 (UTC)
I removed the following passage: "A previous version of this entry contained the following (false) claim: "A regular cardinal is measurable if there is a cardinal preserving generic extension of V in which the cardinal is singular." This is an interesting error, so it deserves a couple of comments:
1. There is a forcing notion, due to Prikry and usually called Prikry forcing, that shows that if a regular cardinal is measurable, then there is an extension of V as claimed. In fact, Prikry showed that if a regular cardinal κ admits a Rowbottom filter, then the version of Prikry forcing for this filter preserves all cardinals while making κ singular. However, not only measurable cardinals, but also, for example, real-valued measurable cardinals, admit such a filter. Real-valued measurable cardinals are not measurable.
2. However, if a cardinal is regular in V and singular in a cardinal preserving extension of the universe, then there is an inner model of V with a measurable cardinal." b/c I don't believe it's Kosher for an article to refer to itself as an article -- if this discussion can be framed in a different way, then it could be reinserted. Zero sharp 20:03, 6 August 2006 (UTC)
- Yeah, I think 0# is right here. I also think the point made by the removed text, while interesting, is not one of the first 2,543 things you'd want to tell someone about measurable cardinals. Let's try to keep a little encyclopedic focus. --Trovatore 21:01, 6 August 2006 (UTC)
