Re: Well ordering of reals

On 26 Jan, 14:11, magi...@xxxxxxxxxxxxxxxxx (Arturo Magidin) wrote:
In article <afc07658-d685-43df-bf17-9b422bfea...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx>,

pers...@xxxxxxxxxxxxxx <pers...@xxxxxxxxxxxxxx> wrote:
On 28 Dec 2008, 03:04, "pers...@xxxxxxxxxxxxxx"
<pers...@xxxxxxxxxxxxxx> wrote:
Hello,
=A0 =A0 =A0 =A0I read somewhere that reals are well ordered, without AC. =
Is it
true?. Any references?. And also a small explanation if possible
please. I am quiet surprised that reals can be well ordered.
Thanks

Well, I have a really really dumb question. The statement "Reals are
well ordered" means that Reals have a minimal element, doesn't it?.

The statement "Reals are well ordered" is either incomplete or
incorrect: either the ordering that makes the reals well-ordered is
missing, or else by implication we are talking about the natural
order, in which case the reals are not well-ordered relative to that
order.

So let's say that we really meant "The reals have a well-ordering, w".

In that case, that means that the reals have a minimUM element,
->relative to w<- (or a w-minimum element).


So, from what you have said -
If "There exists a well ordering, w on reals", then it does mean that
in w, there is some minimal element. That is the set of reals start
with the minimal element and go on.
This is the part that I am struggling with to imagine.

I would feel fine if there exists a well ordering w of Reals in which
every proper subset of Reals has a minimal element. If Reals
themseleves are given a minimal element its a little hard to digest.





(I mean this question has nothing to with AC really, just about the
definition of well ordering.).
Well foundedness I think means any non-empty subset should have a
minimal element.

No, well-foundedness means that every non-empty set has a minimal
element relative to the order relation given by "is an element of".

Not necessarily a proper subset.

Of what?



(Or is it?). So,
Reals should have a minimal element.

The "so" is a non-sequitur. And a minimal element relative to what
ordering?

That is not possible at all.

Huh?

So,
how can you ever say Reals are well ordered.

Relative to what ordering?

(I am looking for the right definition of well ordering.)

The natural order of the real numbers is ->not<- a well-ordering; I
don't think anybody has asserted it is. "The reals can be
well-ordered" is a theorem, that really states that there is an order
relation w on the reals, ->which has nothing necessarily to do with
the natural ordering of the reals<-, which is a well-ordering on the
reals. That is all it says. I don't really understand why you brought
in well-foundedness.

--
======================================================================
"It's not denial. I'm just very selective about
 what I accept as reality."
    --- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.

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