The notation used in this post is so non-standard, I think that you need to define what each symbol means or how it is used.
In Kunen's book, one of the exercises is the following:
Show that for , |H( )| = .
But, H( ) = V( ) and |V( )| = . Also, (because is the cardinality of the union of omega many finite sets (the characteristic functions for each n)).
Is this wrong?
Thanks in advance!
is used as a variable over cardinals.
| | denotes the cardinality of .
is the cardinality of the union of for .
H( ) denotes the set of sets which are hereditarily of cardinality .
V( ) is the level in the cumulative hierarchy.
Hope this helps.
Please say if I've left anything important out.
Just a tangential point: the notation used is not non-standard as far as I am aware. Kunen's book is a standard text, which contains all of this notation except V(\alpha). In Kunen R is used instead of V. In most modern treatments, though, V is used. See Von Neumann universe - Wikipedia, the free encyclopedia, for an example.