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MoeBlee wrote:
> Rupert wrote:
> > If ZF is consistent, then the proposition that the set of functions
> > from the reals to the reals can be totally ordered is indepedent of ZF.
> > Cohen proved this at the same time as he proved the independence of the
> > continuum hypothesis. See "Set Theory and the Continuum Hypothesis".
>
> I had a hunch it was in the Cohen package. Thanks for confirming.
>
> >
> > > And what about "every set can be partially ordered"?
> > >
> >
> > I would think this is provable, consider the transitive closure of the
> > membership relation.
>
> Isn't partial ordering as simple as observing (as I did in my second
> post) that the subset relation is a partial ordering? But I'll think
> about transitive closure too (I need to study that concept anyway).
>
Yes, your proof works as well. Also, {<x,x>|x in X} is always a partial
ordering on X, so the result really is trivial.
> Thanks,
>
> MoeBlee
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