On 12/6/2025 6:31 PM, 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.
*Except that you can't describe how it could be done! That's why we
can apply*
*the AoC in the limited case of a single uncountable set, and the AoC
just says*
*we c**an do it, but doesn't tell us how. AG*
That's wrong. The axiom choice says that given some infinite sets, it
is possible to form a set consisting of one element from each. It
doesn't need to say how because it is just used as an existence proof.
Brent
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