Plus ca change wasRe:Well Ordered Sets



In the following dialogue note the following:

Eric refused to accept that the square root of negative number is
imaginary.
He completely failed to understand the concept of a "well- ordered"
set.
He failed to see that events on a world line are well- ordered.
He resorted to obscenity and insult ("*** me, you're dense")
He had made a mistake and spent weeks denying it:
Finally he acknowledged getting something "backwards" but thought that
it wasn't important
and he shouldn't have been corrected.

This all happened last August, which is a quiet month for me.

This August we've gone through the process all over again with his
misunderstandings of "events" and "worldlines".

Plus ca change...

On Aug 18, 9:36 am, Eric Gisse <jowr...@xxxxxxxxx> wrote:
On Aug 18, 3:54 am, Yuan...@xxxxxxxxx wrote:


Jenny:
If . d\tau^2 = -ds^2 and if d\tau is real, then ds must be
imaginary,

Eric:
You aren't paying attention...ds^2 is LESS THAN ZERO.

Jenny:
When you take the square root of -1, you can choose +i or -i.

Eric:
A complex number cannot be "positive" or "negative" because there is
no well-defined notion of well-ordering. You can't say one complex
number is bigger than another because there are an infinite number of
complex numbers that have the same magnitude.

Jenny:
Well-ordering has little to do with being "bigger" or being positive
or negative.
In the context of ordering, a < b means that a "precedes" b, not that
b is bigger than a.

The concept well ordered does not *only* apply to real numbers.

We can treat any "simple" line in the complex plane as a well ordered
set.

Moreover we can treat the set of events occupied by a particle in
spacetime (its "world line") as a well ordered set.

Eric:
Same problem, bigger boat. Events in space-time are not well-ordered
either.

Jenny:
The concept of being well ordered relates to sets.

In my example that set was the events occupied by a particle in
spacetime (its "world line")

It does not follow that because the set of all events in spacetime is
not well ordered then the set of events occupied by a particle in
spacetime cannot be well ordered.

That might well turn out to be correct in some version of quantum
gravity, but in this context it's a strange position to take.

Eric:
This is a meaningless discussion. You cannot say "Well I'll only
consider the imaginary line y = ix..." because you cannot describe
what i is WITHOUT the rest of the complex plane.

Jenny:
I can say I'll consider only the set black dogs even though I have to
use the concept of "dogs".
And I can consider only the imaginary line (or any line) it as a
particular set in the complex plane.
Just as a well ordered world line is a particular set of events in
space time.

Setting c = 1,
d\tau^2 = -ds^2
Try expressing d\tau in terms of ds.

Eric:
*** me, you are dense. This is the zillionth time now: ds^2 is LESS
THAN ZERO for time-like paths so d\tau^2 is positive. Which is the
whole point of the definition of tau.

Jenny:
Try expressing d\tau in terms of ds.
d\tau = ?

Eric:
Paths parameterized by proper time have ds^2 minimized thus d\tau^2
maximized. Thus, the heart of his intense whining about not being
able to understand is revealed.

Jenny:
You wrote:

"Maximization of ds is minimization of d\tau"
which is transparently wrong,

Eric:
Heh. Yea, it is.
I got it backwards, and he didn't notice.

Jenny:
Well, that's what I was correcting, in my first post - if you trouble
to go back and read it.
Once again, your first response on having a mistake pointed out is to
deny that you made a mistake.

It's day one of complex analysis.

Complex numbers are ordered pairs, (a,b), where in essence the a's are
real and the b's imaginary.
Nevertheless, whether ds is real or imaginary is purely a matter of
convention in the choice of metric.

With the (+ - - -) metric, we have d\tau^2 > 0 for timelike paths.
In which case, ds^2 < 0,. All of which is by convention, as we both
agree.

Eric:
You say it, but you don't appear to believe it.

Jenny:
Of course I do, it's you that have trouble understanding that ds must
be imaginary if ds^2 <0.



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