Is Dedekind's Axiom Still Relevant in Modern Mathematics?

[Canute]

You're quite right. I happened to notice your reply before I logged off. I didn't mean to just have the last word and run off, sorry if that was the impression. I was just explaining my comment about the paradoxicality of motion if spacetime is treated as series of points a la Zeno, which you'd asked about, and explaining that I was about to disappear from the discussion. I didn't want to go without warning. I shouldn't have mentioned Lynds. I'd like to discuss Zeno with you but will have to come back another time. Sorry if it seemed I was point scoring. My mistake.

Regards
Canute

 

[HallsofIvy]

Dedekind's axiom doesn't have anything to do with "motion" or Zeno. It is equivalent to the least upper bound property (or "monotone convergence", or the "Cauchy criterion" or the fact that the set of real numbers is connected (with the usual topology) or that every closed and bounded set of real numbers is compact (again with the usual topology).

 

[mathwonk]

you are not to blame. Dantzig's axiom is poorly worded since as stated it did not define the word "severing".

a more precise statement would have been: "... then either the first class has a largest element or the second class has a smallest one."

Unless this is a very good book for other reasons, I would try to find one that is written more carefully.

 

[Canute]

Aha. That makes sense. It hadn't occurred to me that Dantzig might have misrepresented D's axiom. Thanks.