Given any cardinals and in , we can use the Levy collapse to give a new model where . Let be the set of partial functions with . These functions each give partial information about a function which collapses onto .
Given any generic subset of , has a set , so let . Each element of is a partial function, and they are all compatible, so is a function. since for each the set of such that is dense (given any function without , it is trivial to add , giving a stronger function which includes ). Also since the set of such that is in the range of is again dense (the domain of each is bounded, so if is larger than any element of , is stronger than and includes in its domain).
So is a surjective function from to , and is collapsed in . In addition, , so it satisfies the chain condition, and therefore is not collapsed, and becomes (since for any ordinal between and there is already a surjective function to it from ).
|Date of creation||2013-04-16 22:08:32|
|Last modified on||2013-04-16 22:08:32|
|Last modified by||e1568582 (1000182)|