composition preserves chain condition
We prove that there is some such that any generic subset of including also includes of the . Then, since satisfies the chain condition, two of the corresponding must be compatible. Then, since is directed, there is some stronger than any of these which forces this to be true, and therefore makes two elements of compatible.
Claim: There is some such that
(Note: , hence )
If no forces this then every forces that it is not true, and therefore . Since is regular, this means that for any generic , is bounded. For each , let be the least such that implies that there is some such that . Define for some .
If then there is some such that , and if then must be incompatible with . Since satisfies the chain condition, it follows that .
Since is regular, . But obviously . This is a contradiction, so we conclude that there must be some such that .
If is any generic subset containing then must have cardinality . Since satisfies the chain condition, there exist such that and there is some such that . Then since is directed, there is some such that and . So .
|Title||composition preserves chain condition|
|Date of creation||2013-03-22 12:54:40|
|Last modified on||2013-03-22 12:54:40|
|Last modified by||Henry (455)|