Cardinality of proper classes:
zuhair
Is there a definition of Cardinality of proper classes.
I know of one example were that can be defined, and that is when
we have global choice, so in MK\NBG with Global choice, we can
define Cardinality in usual manner using Von Neumann Cardinals.
However what about defining Cardinality of proper classes when we
don't have Choice?
One of the strange results that I've noticed was regarding my
attempts to define Cardinality of *set*s using the concept of
hereditarily sets.
I defined Cardinality of *set*s as:
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For every set x:
Card(x) is the set of all sets Equinumerous to x, having every
member of their transitive closures strictly subnumerous to x.
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This definition works in ZFC.
Also it works with theories weaker than ZFC, like
ZF+ for all x Exist y (Card(x)=y & ~y=0).
However lets modify this definition to the following:
For every *class* x:
Card(x) is the class of all sets hereditarily strictly
subnumerous to x.
In symbols:
Card(x)={y| for all z (z e TC({y}) -> z is strictly subnumerous to x)}
Using this definition restricted to *sets* works in ZFC.
Also it can work in weaker theories like
ZF+ for all x Exist y (Card(x)=y & x subnumerous to y).
However my interest here is regarding using this definition to define
cardinalities of proper classes as well.
The strange result is that if we work in MK\NBG with choice only,
i.e. without global choice, then this definition will yield incorrect
results, since the cardinality of the class of all set ordinals would
always be equal to the cardinality of the class of all sets, thus
implicating global choice, which is not necessarily the case.
However this definition works with global choice nicely, and it can
define cardinalities of all classes weather they are sets or proper
classes.
The strange matter is that without choice this definition can have
some interesting results, it can tell us that the cardinality of the
class of all set ordinals is lesser than that of the class of all
sets, so it can differentiate between cardinalities of some proper
classes that are not equinumerous to each other, but I don't know if
it does that always.
However still this definition doesn't work nicely to be acceptable to
be a definition of cardinality of proper classes (except when global
choice is assumed).
Now what are the known trials to *define* cardinality of proper
classes.
By the way the *primitive* concept of cardinality can be used to
deal with this issue, but my question was if a defined concept of
cardinality of proper classes in MK\NBG is possible? and what are
the known trials in this issue.
Zuhair