Re: all sets are aleph_2 ordable

On Mar 12, 9:20 am, amy666 <tommy1...@xxxxxxxxxxx> wrote:
by tommy1729's request , his words :
all sets are aleph_2 ordable.
meaning that all set are well ordered if we can order
aleph_2 sets well.
and thus all sets are well ordered if aleph_2 is well
ordered.
and then if and only if aleph_2 is well ordered ,
then the axiom of choice holds.

OK, I think I know what's going on here.

We already know that tommy1729 is a fan of Pocket
Set Theory. In PST, all sets are countable, and
any uncountable class is a proper class. It is also
an axiom of PST that there exists a (class) bijection
between any two proper classes. One then proves that
the universe has cardinality aleph_1, and that the
universe can be wellordered -- since there exists a
bijection between the universe and the (wellordered)
class of all ordinals. In other words, Global Choice
is a theorem of PST.

Now tommy1729 has his own theory, TST, which is
similar to PST, except that the universe has
cardinality aleph_2 rather than aleph_1. What
tommy_1729 is asking is whether Global Choice would
follow in his own theory.

In other words, if we have a class theory in which
the universe has cardinality aleph_2, does Global
Choice necessarily follow?

Notice that tommy_1729 believes that it does. I
believe that someone, in another thread, proves
that Global Choice does _not_ follow.

For example, consider ZF-Powerset. The only
problem here is that ZF officially has no proper
classes, so we can't really use them. (NBG does
not work here, since NBG has an Axiom of Size
Limitation which _does_ imply Global Choice.)

Then the class of all ordinals has cardinality
aleph_1, but we can't prove that the universe
has cardinality aleph_1. We need something like
PST's "all proper classes are the same size" (or
NBG's Size Limitation) in order to prove Global
Choice as per tommy1729's wishes.
.

Relevant Pages

  • Re: Proper classes
    ... "domain" and "codomain" are allowed to be proper classes, my recollection is that the concept is not particularly interesting from the viewpoint of cardinality, since all proper classes would turn out to have the same "cardinality" in this extended sense. ... Without global choice you can have incomparable proper classes, as can be seen rather trivially: take a model of ZFC in which there is no definable well-ordering of V and add on top of it the definable subsets as classes and you'll have a model of NBG in which there is no bijection between V and the class of ordinals. ...
    (sci.math)
  • Re: Sets and classes.
    ... cardinality, hence the full class is too big to be a set. ... A collection of sets with unbounded cardinality is a level 2 class. ... Then, if I got it right, X is a "Grothendieck universe". ... The cardinality of a Grothendieck universe is always a strongly ...
    (sci.math)
  • Re: Theories M,
    ... is the set of all sets hereditarily less than cardinality kappa. ... hereditarily countable sets. ... Let U_0 be the smallest universe, ... But notice that U_omega _can't_ have a predecessor universe. ...
    (sci.logic)
  • Re: Theories M,
    ... Theories M are a constilation of set theories, ... is the set of all sets hereditarily less than cardinality kappa. ... Axiom 5) guarantees that "e" is identical ... Let U_0 be the smallest universe, ...
    (sci.logic)
  • Re: Theories M,
    ... is the set of all sets hereditarily less than cardinality kappa. ... Axiom 5) guarantees that "e" is identical ... Let U_0 be the smallest universe, ... But notice that U_omega _can't_ have a predecessor universe. ...
    (sci.logic)