FOM: V=L and large sets

Colin McLarty cxm7 at
Fri Apr 28 08:58:10 EDT 2000

	The axiom candidate V=L is incompatible with large cardinals, but entirely
compatible with sets as big as large cardinals. All the cardinals in any
ZFC universe are constructible there, of course.

	Is there a way to state in ZFC that "there exists an ordinal big enough to
be measurable" without deciding whether V=L? Or for other largeness

	For example, I believe you can say in ZFC: "o is an ordinal and the set
V(o) has a forcing extension which models ZFC+(there is a measurable)". If
such an ordinal exists, it is much more than "big enough" to be measurable.
Yet it seems that positing such an ordinal, will still be consistent with
assuming V=L. Or am I somehow wrong about that? 

	Is there some much simpler way to describe ordinals "big enough" to have
various large cardinal properties?

thanks, Colin

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