Re: necessary or sufficient properties for 'greater than' ( >)

On Fri, 2 Jan 2009 09:39:44 -0800 (PST), nukeymusic
<NuKeyMusic@xxxxxxxxx> wrote:

On Jan 2, 5:25 pm, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
On Fri, 2 Jan 2009 08:04:08 -0800 (PST), nukeymusic



<NuKeyMu...@xxxxxxxxx> wrote:
On Jan 2, 4:39 pm, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
On Fri, 2 Jan 2009 06:29:24 -0800 (PST), nukeymusic

<NuKeyMu...@xxxxxxxxx> wrote:
On Jan 2, 3:02 pm, "G. A. Edgar" <ed...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>
wrote:
In article
<fa5d5b1b-80ea-45af-88bd-c0d12c765...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,

nukeymusic <NuKeyMu...@xxxxxxxxx> wrote:
I wonder which properties are necessary or sufficient to define
correctly -i.e. in a way which makes sense mathematically- 'greater
than' for a given set. I have been searching a bit already on the
Internet and thus far I came to the following conclusion (but I'm
rather unsure about it):

1. not (a<a)
2. if a<b then not(b<a)
3. if a<b and b<c then a<c
4. if a<b and c>0 then ac<bc

Your 'given set' has multiplication? and 0? Why not addition?

The reason why I added the 4th condition (which indeeds implies a 0
and multiplication for the given set) was that the lexicographic order
of the complex numbers as explained onhttp://www.cut-the-knot.org/do_you_know/complex_compare.shtml
makes that order "not useful", I thought this fourth condition might
be generally necessary or maybe a consequence from another condition
(unknown to me) which is universally considered to be necessary to
make sense of ">"
So my question remains: is there a universal "4th condition"? If not
how then can I check whether a certain way of ordening a set makes
sense?

First you need to _define_ _exactly_ what you mean by "makes
sense".
I thought (or hoped) there was a consensus amongst mathematicians
about what "makes sense" in this context and wanted to learn about it.
As I mentioned before condition 4 seems to make the lexicographic
order of complex numbers doesn't make sense or to make it useless. You
are probably right it's better to narrow the question of "making
sense" to the set of complex numbers first. I know the standard answer
is that the complex numbers can't be ordered in a way that 'makes
sense' or is useful but I really am interested to know against which
criteria any attempt to order them should be checked. (I hope my
English is understandable, I am not a native speaker)

It's simply not true that the lexicographic order on C "doesn't
make sense". It makes perfect sense. Whether or not it's useful
depends on what you're trying to do. It's not true that the complex
numbers cannot be ordered in a way that makes sense.

You say that "the complex numbers cannot be ordered in a way
that makes sense" is "standard". It's certainly not. But it gives
a hint regarding what it is you may be misremembering:
Even though it was years ago when I first learned about complex
numbers I remember very well this was what we were taught then. I even
remember the teacher made an unconvincing attempt to make me accept
that "truth" saying that comparing complex numbers was like comparing
the color of a mix of paints in terms of the main colors...
There is no order on C that makes C into an _ordered field_.
Thanks for pointing to the right "terminology", I appreciate it very
much. Do you know about a simple proof of "There is no order on C that
makes C into an _ordered field"

Read the rest of my post! The proof is very simple. First you
need the _definition_ of "ordered field", which you can find
at the Wikipedia page mentioned below. The proof is also
given there.

(I tried out a few ways to order C and
all of them indeed work out the wrong way but that is not a proof of
course).

best regards and thanks for all the help offered so far
nukey

See

http://en.wikipedia.org/wiki/Ordered_field

for the definition, and also for an answer to the question of
which _fields_ can be given an order so as to become an
ordered field.



Any help appreciated
thanks and best regards,
nukey

No, maybe that's second. First you need to say exactly what sort
of additional structure you're assuming - you start by saying
S is just a set, but your condition (4) makes no sense, given
only that S is a set.

thanks and best regards,
nukey

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

Relevant Pages

  • Re: not anymore ?
    ... "Understanding Godel isn't about following his formal proof. ... That would make a mockery of everything Godel was up to." ... (John Jones, "My talk about Godel to the post-grads." ...
    (sci.math)
  • Re: -- g o g = f
    ... that f is 1-1 then it's nothing but simple set theory. ... That would make a mockery of everything Godel was up to." ... (John Jones, "My talk about Godel to the post-grads." ...
    (sci.math)
  • Re: 3 functional equations is max
    ... an explicit formal definition of "independent". ... That would make a mockery of everything Godel was up ... (John Jones, "My talk about Godel to the post-grads." ...
    (sci.math)
  • Re: why does professor david c ullrich have to put people down to feel good about himself?
    ... "Understanding Godel isn't about following his formal ... (John Jones, "My talk about Godel to the post-grads." ... That would make a mockery of everything Godel was up to." ...
    (sci.math)
  • Re: Question on asymptotics
    ... "Understanding Godel isn't about following his formal proof. ... That would make a mockery of everything Godel was up to." ... (John Jones, "My talk about Godel to the post-grads." ...
    (sci.math)