Re:What is the Axiom of Choice?
by jfengel on Friday July 16, 2021 @01:46PM
(#27858977)
It's not just about ordering, but well-ordering. Which says not just that you can put them in order, but that you can put them in order with a first element (one that's always less than all of the others).
So you can have a set like "all real numbers greater than 0". It doesn't have an obvious first element, even though you can always compare any two numbers and say "this one is less than that one".
There may be a non-obvious first element, i.e. some strategy for picking a first element that applies to all sets. That's the Well-Ordering Principle. Which feels false to most mathematicians, but it's equivalent to the Axiom of Choice, which feels true.
So you can have a set like "all real numbers greater than 0". It doesn't have an obvious first element, even though you can always compare any two numbers and say "this one is less than that one".
There may be a non-obvious first element, i.e. some strategy for picking a first element that applies to all sets. That's the Well-Ordering Principle. Which feels false to most mathematicians, but it's equivalent to the Axiom of Choice, which feels true.