Re: Cantor Confusion

On Fri, 12 Jan 2007 13:33:44 EST, Andy Smith wrote:

The crossings are totally ordered but they are not
well ordered, nor is
the reverse ordering well ordered.


How does one define "well ordered" as distinct from
"totally ordered", please?

A set is totally ordered if it satisfies the
trichotomy law. That is, if
x and y are members of the set, then exactly one of
the following
statements holds:

(1) x < y,
(2) x > y,
(3) x = y.

OK, fair enough. (how could a pair of elements be otherwise
if there as a well defined function f(x) of each
element
on which the elements are sorted ?).

And what if there isn't? For example, consider the set of all subsets of
N, ordered by inclusion. This is a partial order. It's not a total
order, because it doesn't satisfy the trichotomy law. For example, if E
= set of even numbers and P = set of primes, then neither is a subset of
the other and they are not equal.


A set is well ordered if every nonempty subset has a
least member. Your
set of crossings is not well ordered because the
subset of crossings for
x > 0 does not have a least element.

But presumably it is acceptable if every nonempty
subset does not have a greatest member? In the context
of this discussion, that is a bit circular, isn't it?

The definition of a well ordering does not say anything about greatest
members. How can failure to mention something possibly make a definition
circular?

I am sure that it is incompatible with Peano to have
a countably infinite ordered set with finite boundaries.

Then you are wrong, because the rationals in [0,1] are such a set.

But the set of zero crossings exist as an ordered structure
as a counter example. Of course you can pull out your (carefully
self consistent) rule book & rule it off-side, but that is
maybe the point?

When have I ever said there was anything wrong with your set of
crossings? You are simply not listening. If I say a red ball is not
blue, am I finding fault with the ball?

Every ordinal is well ordered. Each natural number,
being a finite
ordinal, is likewise well ordered. The set of all
natural numbers is
identical to the ordinal w and is therefore well
ordered. Every well
ordered set is necessarily totally ordered.

Some examples of totally ordered sets that are not
well ordered are:

(1) Your set of crossings.
(2) The reals in [0,1].
(3) The rationals in [0,1].
(4) The integers.

The fact that two ordered sets have the same
cardinality does not imply
that they have the same order type. You need to
describe more carefully
how you are associating those two sets, keeping in
mind that each bit
position of an integer is finite.

Well the point I was suggesting was that the
infinite sequence
of crossings is an ordered set, with a start and an
end
and even if one can't label each term with an
ascending
integer, their locations exist on the real line as
much as any other fairyland construction. (and,
hence, maybe, other
infinite sequences e.g. 11...11 with a beginning
and an end might
exist and have some meaning?)

Certainly the crossings exist. That has never been
in dispute.

You are free to write digit sequences such as
11...11, but they have no
particular meaning as numbers until you explain what
you mean by them.

What can you say then about an ordered countably
infinite set that
has a well defined beginning and an end - or does
the fact that it
have a beginning and an end automatically
disqualify
it from being a legitimate object?

Anything that can be rigorously defined is a
legitimate mathematical
object. The set w+1 is an ordered countably infinite
set that has a well
defined beginning and an end.

But not a well defined end?

When I say something has an end, that means it has a well defined end.
What other kind of end could it have? The last element of w+1 is w.

Like pointing to e.g.
some definition of a number on the real line, and saying
"there it is". Can you give any concrete example of
an actually realised set of transfinite numbers? (bearing
in mind that I am an earthy physicist, I want some
explicit points that I can see on a calculator, making up
the transfinite set).

Can you give any concrete example of an actually realised number 3? I
don't mean an example of three objects, but an instance of the actual
number 3.

I don't buy the set of all ratioanls in [0,1] as meeting
my criteria of a well ordered infinite sequence with
finite boundaries, back to 3 wishes again. The set of
rationals is not acceptable because there is no last member
as a predecessor to 1 (whereas, with my zero crossings,

All that proves is that you failed to ask for what you really wanted. We
have been talking in circles because you have never specified just what
it was that you wanted to know.

Perhaps this will help: we say a set X is *finite* if X can be well
ordered in such a way that each nonempty subset of X has a greatest
element.

A different way of saying it is that a set X is finite if it can be
placed in bijection with some natural number. It's fairly easy to show
by induction that these two notions of finiteness are equivalent.

there exists a last one, its predecessor, that ones predecessor, etc ...)



--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
.

Relevant Pages

  • Re: Cantor Confusion
    ... set of crossings is not well ordered because the ... subset does not have a greatest member? ... a countably infinite ordered set with finite boundaries. ...
    (sci.math)
  • Re: Cantor Confusion
    ... infinite ordered set may have a last member. ... that sense a sequence ... Each of the crossings is at some finite x. ...
    (sci.math)
  • Re: Dial 999 for the real number line
    ... conclude that there is a set that has no largest member. ... And to distinguish two infinite decimals, ... two reals is always finite. ...
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  • Re: Logarithm of transfinite numbers
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  • Re: Orlow cardinality question
    ... > to be finite if the set of naturals up to and including it is a finite ... > set by the Cantor definition, is, by the Cantor definition, an infinite ... > member of the set. ... Even worse, I posit that the first may come after the last, and the last before ...
    (sci.math)
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