Re: A simple question?

Jules wrote:
Actually, that quote
For example, the real numbers are not well
ordered, but there is a well-ordering of the real numbers which is not
the standard ordering.
came from my post. I meant that such an ordering exists, if one
assumes the axiom of choice. I meant to use this simply as an example
that a set need not be well ordered, even if a well-ordering of the set
exists.

What does it mean for set not to be well ordered even though a well
ordering of the set exists?

A better example is probably the set Z of integers. Z is
certainly not well-ordered (it has no minimal element),

The standard ordering is not a well ordering, but that does not entail
that the set is not well ordered. Maybe this is just a difference in
how we use these phrases, but I say a set is well ordered iff there
exists a well ordering of the set. So the fact that the standard
ordering is not a well ordering doesn't contradict that the set is
nonetheless well ordered.

but a
well-ordering of Z can be explicitly described.

Okay.

I still want to know
what the "natural" ordering of P(w) is.

My impression is that what is being said in this thread is that there
is no such thing as "the natural ordering ot the power set of omega".

MoeBlee

.

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