Re: A simple question?

MoeBlee wrote:
Jules wrote:
MoeBlee wrote:
zuhair wrote:
Hi,

For w={ 0,1,2,3,....} , is P(w) a well ordered set?

You mean the power set of omega?

Your question is not complete.

Under what axioms?

A set is well ordered if there is a well ordering of the set.

I do not think this is correct. Well-ordered can only apply to a
totally ordered set.

I don't understand your objection. Every well ordering is a total
ordering (but, of course, not every total ordering is a well ordering).
If there is a well ordering of a set then that well ordering is also a
total ordering of the set. When we say that a set is well ordered, we
just mean there is a well ordering of the set. Or, as I mentioned, in
some instances, we can be more specific and say that a certain ordering
is a well ordering of the set.

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.

No, in ZFC, the set of real numbers is well ordered, since there is a
well ordering of the set of real numbers. The fact that the standard
ordering of the real numbers is not a well ordering is a separate
issue. When we say that a set is well ordered, we don't not necessarily
claim that the standard ordering is the well ordering.

Halmos says, "A partially ordered set is called *well ordered* (and its
ordering is called a *well ordering*) if every non-empty subset of it
has a smallest element."

--
David Marcus
.

Relevant Pages

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  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
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