Re: A simple question?

MoeBlee wrote:
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.

To me, if someone says a set is well-ordered, they mean whatever
ordering relation the set has (or whatever ordering relation we are
talking about) is a well-ordering. It doesn't mean that I can construct
a new ordering relation that is a well-ordering.

Do you see somewhere in Enderton where he uses "well-ordered" to mean a
well-ordering exists?

--
David Marcus
.

Relevant Pages

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