Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
From: Timothy Little (tim-via-n.i.net_at_little-possums.net)
Date: 01/21/05
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Date: 21 Jan 2005 02:35:25 GMT
Ross A. Finlayson wrote:
> In ZFC, there is at least one well-ordering of the reals.
As we both agree, and doesn't need to be repeated every time.
> A well-ordering is necessarily a total ordering so the
> well-ordering(s) are a subset of the total ordering(s).
Yes. More pointless repetition.
> I'd like to know more about the well-orderings of the reals because
> for any well-ordering of the reals, it is a well-ordering of each
> subset of the reals, and thus, for example, a well-ordering of the
> rationals.
I'd like to know more about it too.
> One of the points I want to make clear to you is that it's possible
> to consider that the only total ordering of the reals that is a
> well-ordering is a monotonic function from the naturals to each
> element of an interval of the reals, the normal or "usual" ordering.
It's possible to consider, yes. As in, wonder what consequences
follow and whether it might be true or false. However, reasonable
people stop considering it when it is shown to be self-contradictory.
> Here, the Cartesian product X x Y is indicating a function from Y to
> X.
You're confused. There is a relationship between a bijection X<->Y,
and some *subset* of X x Y, but the cartesian product does not
indicate a function.
> As the reals are restricted to the unit interval, then R_n is some
> infinite subset of the unit interval that is well-orderable because of
> a bijection between it and N, and because R_n is a subset of R and N is
> a subset of O.
There's no guarantee that any well-ordering of R will map N into the
unit interval, but there does exist a well-ordering for which this
property is true.
> So, when there is a well-ordering of the unit interval then some subset
> of the unit interval is well-ordered by a function between that set and
> the natural integers. That could be f(n)=1/n
It could indeed.
> Indeed, from what I understand of the Pi and Sigma Goedelian
> complexity definitions a well-ordering of the reals would be some
> random scattering of points, in terms of the normal ordering.
Yes, very likely.
> While that's so, with using the normal ordering it's possible that
> R_n = R, where infinite sets are equivalent.
In ZFC, infinite sets are not equivalent. If you want to devise a
system of mathematics in which they are, go right ahead. Just don't
expect anyone but you to be interested in it. I'm here talking about
ZFC, as it seemed you were too.
"Ross A. Finlayson wrote:
> In ZFC, there is at least one well-ordering of the reals."
> the result among the participants that the normal ordering of the reals
> is _not_ a well-ordering.
At least we're making some sort of progress.
> So, basically I'm taking a theory that defines the numerical system
> up to the reals and then adding that the normal ordering is a
> well-ordering
Which theory? So far, you've been talking about ZFC. If you are
really talking about some other theory, you'll have to present or at
least give references to the axioms and inference rules of that
theory.
In ZFC, if you add "the normal ordering is a well-ordering" to the
definition of the reals, then you get a trivial system that states
that every well-formed mathematical statement, even contradictory
ones, is true. I don't think you want that.
> To that end, I wonder what I should avoid.
Don't include the rationals in the set you want to well-order with
their usual ordering. The trouble is, once you do that you end up
with something that doesn't look anything like a good model for a
geometric line.
In fact, it is pretty easily shown that if every element in some set
has a "previous" point in some order, that order can't be a
well-order.
- Tim
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