
Re: Van Neumann Hierarchy
Posted:
Apr 27, 2014 12:11 PM


On Sat, 26 Apr 2014 20:01:26 +0100, Sandy <sandy@hotmail.invalid> wrote:
>dullrich@sprynet.com wrote: >> On Fri, 25 Apr 2014 20:13:43 0700, William Elliot <marsh@panix.com> >> wrote: >> >>> The Van Neumann hierarchy is defined with transfinite induction as >>> V_eta = \/{ P(V_xi)  xi < eta }. >> >> Huh> That reminds me of the Von Neumann hierarchy... > >May I ask how the case eta = 0 is to be interpreted? My vague thoughts: >There is no xi < 0 so V_xi can be assigned no meaning. So P(V_xi) is >also without meaning, and \/{etc} is without meaning. > >The version of the definition of von Neumann's hierachy that I have seen is: > > V_0 = 0
Where 0 is the empty set...
> V_{alpha+1} = P(V_alpha) > V_{lamba} = U_{alpha < lambda} V_alpha (lambda a limit ordinal). > >I have not seen a oneclause definition before reading William Elliot's >post.
What he wrote is ok, at least in that regard.
Officially in set theory a single set has a union: If S is a set then U S is {x : x in y for some y in S}.
It follows that U 0 = 0. Since as you point out there is no xi < 0 we have
V_0 = U {V_xi : xi < 0} = U 0 = 0.
> > >> >>> >>> Within ZF, using regularity and replacement, how does one show >>> that for every set A, there's some ordinal eta, with A in V_eta. >>> >>> It would suffice to show for every transitive set A, >>> there's some ordinal eta, with A in V_eta. >>> >>> With that, since every set is a subset of it's transitive >> >> You really really really need to stop criticizing others' English. >> Or learn a little English yourself. It's "its", not "it's". >> >>> closure, every set would be inside the Van Neumann hierarchy. >>

