Why is "0# exists" independent?

[Timothy Y. Chow]

Martin Dowd wrote:
> Thus, there seems to be a "hole" in ZFC, which leaves the question of 
> whether 0# exists completely up in the air. My question is, why does 
> this hole exists? Why does our understanding of the cumulative hierarchy 
> fail to shed any light on the question?

One can also ask, why doesn't ZFC settle "V=L" one way or the other?

Is there some reason you're asking about "0# exists" as opposed to "V=L"? 
It would seem that the two questions raise similar philosophical issues, 
but isn't the independence of "V=L" an even more basic question?

Tim