Re: Every set can be ... ordered?


MoeBlee wrote:
For Z or ZF, "every set can be well ordered" is equivalent to the axiom
of choice.

But what about "every set can be totally ordered"? Is it a theorem of Z
or a theorem of ZF (if so, what is the proof?) or is it independent of
Z and independent of ZF (and who proved it such?).


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".

And what about "every set can be partially ordered"?


I would think this is provable, consider the transitive closure of the
membership relation.

MoeBlee

.

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