2 ms·
How do you feel about well ordering the reals? Because the Axiom of Choice gives you that too and so much more. I'm not a zealot one way or another, but don't a
by Tloewald 13y ago
How do you feel about well ordering the reals? Because the Axiom of Choice gives you that too and so much more. I'm not a zealot one way or another, but don't argue for it based on intuition, it plays both ways.
- ColinWright 13y agoAs the old joke goes: We all know that the Axiom of Choice is true, Well-Ordering of the Reals is false, and who knows about Zorn's Lemma. I use AC when I need it, avoid it when I can do without it, and sometimes look at what it allows versus what it implies.
- thaumasiotes 13y agoI've never really understood why the well-ordering theorem is the version of the axiom of choice that's supposed to be "obviously false". It's pretty trivial to see how you get it from the AC, and since the AC is "obviously true" I've always felt the well-ordering theorem is intuitive as well. (Actually, my mental model of the process generally leads me to want to think "all sets are countable"... c'est la vie :/ ) The Banach-Tarski theorem, on the other hand, is much more robustly "obviously false".