On Saturday, December 6, 2025 at 8:11:37 PM UTC-7 Russell Standish wrote:
On Sat, Dec 06, 2025 at 06:31:20PM -0800, 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 can do it, but doesn't tell us how. AG It is in a sense entirely arbitrary, so any method would do. For example, with the Earth-Moon system, choosing the barycentre suffices, and has a lot of advantages. How would you choose an origin for R^n? AG ---------------------------------------------------------------------------- Dr Russell Standish Phone 0425 253119 (mobile) Principal, High Performance Coders [email protected] http://www.hpcoders.com.au ---------------------------------------------------------------------------- -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion visit https://groups.google.com/d/msgid/everything-list/60022ef7-a994-4a41-96e3-370761218bban%40googlegroups.com.
