On Wednesday, December 3, 2025 at 9:25:18 PM UTC-7 Brent Meeker wrote:
On 12/3/2025 4:10 PM, Alan Grayson wrote: On Wednesday, December 3, 2025 at 2:37:35 PM UTC-7 Brent Meeker wrote: On 12/3/2025 3:00 AM, Alan Grayson wrote: On Tuesday, December 2, 2025 at 3:51:45 PM UTC-7 Brent Meeker wrote: 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 Presumably, you have a more rigorous approach, say for R^2, by imagining a plane, or using a piece of flat paper, and throwing a dart at it to define the origin. AG You can choose any point and make it the origin, which just means you give it the label (0,0). No physics can depend on the choice of origin or the coordinate system. Of course, but how can one find any point on a thing which is ill-defined in the first place? You don't *find *a point on a thing. The coordinate system and the origin are in mathematics, not in reality. *Then why do we need the Axiom of Choice? AG* That's why they can be arbitrary. You just postulate a coordinate system and say I'll call the center of the Earth the origin and z-axis to be the line toward Polaris. No one can say, "That's wrong." because it's arbitrary. Brent -- 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/fe65cbd6-8edd-452a-a62e-1e09f3d4b85en%40googlegroups.com.
