bo198214 Wrote:...

DirkU should be the set theory specialist,

but he is currently on vacation, and my knowledge of set theory is probably even less than yours ...

How nice to be called a set theory expert ... To say something to the question (as I understood it) I have first something to say to a method of set-theoretical independence proofs.

That

is not provable in ZF is usually shown by a forcing construction. There are some variants to do so. Often you will start with a ZF model

and construct another one--say

-- from

which is forced to have some desired properties--for instance that the negation

holds in

of a statement

you want to prove to be independent of ZF. If this is possible then

cannot proved in ZF.

Now the standard construction of the model

to disprove

in ZF is done in way to move

upwards in the hierarchy of cardinal numbers. When you analyze this construction you can see that you can make

make nearly arbitrary large, i.e. you can choose an ordinal number

much greater than 2 and force that

in

. Thus for every ordinal number

such that another such forceable ordinal number

as just described with

exists it is unprovable in ZF that

.