On 12/2/2025 1:24 AM, Alan Grayson wrote:
On Monday, December 1, 2025 at 10:37:16 PM UTC-7 Russell Standish wrote:
On Mon, Dec 01, 2025 at 08:07:14PM -0800, Alan Grayson wrote:
>
>
> On Monday, December 1, 2025 at 3:46:40 PM UTC-7 Russell Standish
wrote:
>
> On Sat, Nov 29, 2025 at 11:13:05PM -0800, Alan Grayson wrote:
> >
> >
> > On Friday, November 28, 2025 at 3:26:03 PM UTC-7 Russell
Standish wrote:
> >
> > Sorry - I can't make sense of your question.
> >
> >
> > The Axiom of Choice (AoC) asserts that given an uncountable
set of sets,
> each
> > one being
> > uncountable, there is a set composed of one element of each
set of the
> > uncountable set
> > of sets. The AoC doesn't tell us how such a set is
constructed, only that
> we
> > can assume it
> > exists. So, in chosing an origin for the coordinate system for
a plane
> say, we
> > have to apply
> > the AoC for a single uncountable set, the plane. But there's
no way to
> > construct it. Does
> > this make sense? AG
> >
>
> I don't see the axiom of choice has much bearing here. To choose an
> origin, we simply need to choose one point from a single
uncountable
> set of points. We label finite sets of points all the time -
geometry
> would be impossible otherwise - consider triangles with vertices
> labelled A,B and C.
>
>
> You write "we simply need to choose one point from a single
uncountable set
> points", but how exactly can we do that! That's the issue, the
construction of
> the coordinate system. In fact, there's no credible procedure
for doing that,
> so
> we need the AoC to assert that it can be done. IMO, this is an
esoteric issue.
> For example, we can't just assert we can use the number ZERO to
construct
> the real line, since with ZERO we have, in effect, a coordinate
system.AG
>
Rubbish - it is not controversial to pick a set of points from a
finite set of uncountable sets. As I said, we've been doing that
since
building ziggurats on the Mesopotamian plain. AoC is only
controversial when it comes to uncountable sets of uncountable sets.
*It's subtle, maybe too subtle for you to see its relevance. You're
imaginIng throwing*
*a dart at a flat piece of paper, but that falls far short of a
viable/construction /of a *
*coordinate system on a plane. You can imagine it being done and
that's the extent*
*of your proof. AG *
A coordinate system is anything that supplies a unique set of numbers to
label every point such that the numbers are continuous. I think you have
and exaggerated idea of what needs to be constructed.
Brent
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