FS iterated forcing preserves chain condition
is the empty set.
Suppose is a limit ordinal and satisfies the chain condition for all . Let be a subset of of size . The domains of the elements of form finite subsets of , so if then these are bounded, and by the inductive hypothesis, two of them are compatible.
Otherwise, if , let be an increasing sequence of ordinals cofinal in . Then for any there is some such that . Since is regular and this is a partition of into fewer than pieces, one piece must have size , that is, there is some such that for values of , and so is a set of conditions of size contained in , and therefore contains compatible members by the induction hypothesis.
Finally, if , let be a strictly increasing, continuous sequence cofinal in . Then for every there is some such that . When is a limit ordinal, since is continuous, there is also (since is finite) some such that . Consider the set of elements such that is a limit ordinal and for any , . This is a club, so by Fodor’s lemma there is some such that is stationary.
For each such that , consider . There are of these, all members of , so two of them must be compatible, and hence those two are also compatible in .
|Title||FS iterated forcing preserves chain condition|
|Date of creation||2013-03-22 12:57:14|
|Last modified on||2013-03-22 12:57:14|
|Last modified by||Henry (455)|