Why is "0# exists" independent?

[martdowd at aol.com]

FOM:

An interesting question occurred to me about 0#; I'ld be interested in any
comments from other participants in the FOM discussion group.  As far as we
know it is not provable in ZFC that 0# does not exist.  If it does, this
has consequences for fairly low levels of the cumulative hierarchy.  None of
these consequences have been shown to hold without at least assuming
that $\omega$-Erdos cardinals exist.

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?

Martin Dowd

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