iterated forcing and composition
Let . This is obviously a member of , since it is a partial function from (and if the domain of is less than then so is the domain of ), if then obviously applied to satisfies the definition of iterated forcing (since does), and if then the definition is satisfied since is a name in for a member of .
is order preserving, since if , all the appropriate characteristics of a function carry over to the image, and (by the definition of in ).
If and are incomparable then either and are incomparable, in which case whatever prevents them from being compared applies to their images as well, or and aren’t compared appropriately, in which case again this prevents the images from being compared.
Finally, let be any element of . Then . If then this is just , and for any . If then . Hence is dense in , and so these are equivalent.
|Title||iterated forcing and composition|
|Date of creation||2013-03-22 12:54:51|
|Last modified on||2013-03-22 12:54:51|
|Last modified by||Henry (455)|