On Saturday, December 6, 2025 at 8:57:49 PM UTC-7 Brent Meeker wrote:
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 It's NOT a proof of anything. That's why it's called an axiom. I am just extending its use to a single uncountable set. AG -- 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/c5a51c0e-2ddd-44f3-ad27-14b818c7861cn%40googlegroups.com.
