Moderator Warning
I just noticed this one. Cognitive Tide, the quickest way to get booted off this site is threaten legal action. You’re gone.
Colibri
General Questions Moderator
CT, the mods have just asked you to dial it back. You do yourself no favours by these kind of comments. the mods do a good job here of keeping discussions such as these civil. the one over-riding principle is, “Don’t be a jerk.”
Many of your comments and your tone in general might be construed to be close to the line.
Welp…
You drive home and this happens.
Ninja’d (And quite convincingly!)
Well, in the very end, he was at least half right…
This is why we can’t have nice things(in GQ). Closed.
samclem, moderator
After re-consideration, re-opened without the flotsam…Or was it jetsam? I always get those mixed up.
Sweet Jebus, samclem, you couldn’t leave well enough alone? 
Step right up, folks! It’s your turn to bring your ignorance and stubbornness out into the open. And if you try you can get banned before school starts. I’ll start you off:
It’s as plain as the nose on your face that .999… = 1. Prove me wrong, either mathematically or with photographic proof that you have no nose.
If only I was Michael Jackson right now…
Can I still have direction with a zero magnitude vector?
From Wiki:
Technically speaking, I think we were actually dealing with derelict
You really don’t get it do you?
Reported.
I used to believe that 0.9999… and 1 were different numbers, but I have read many proofs here and in a few other places that have convinced me they are actually 2 different ways of writing exactly the same number.
I have a hard time getting my head around the Banach-Tarski Theoremif that helps…
Yeah, that Axiom of Choice is a tricky wicket indeed.
The system works.
Personally, I think the Axiom of Choice is incredibly counterintuitive even before you get to outright absurdities like Banach-Tarski.
Mathematician: The Axiom of Choice says that I can pick something from each of these sets.
Me: OK, go ahead, pick something.
Mathematician: Done.
Me: So, what did you pick?
Mathematician: …uh, I can’t say.
Me: You can’t? Do you know which one you picked? How can you call that choosing something?
I mean, if someone told me to choose what to have for dinner, they’d be awfully annoyed at me if all I answered was “I call the dinner I chose X”.
On the other hand, it follows from or is equivalent to a variety of seemingly obvious statements: Zorn’s lemma, the result that every vector space has a basis, Tychonoff’s theorem, etc. The Banach-Tarski paradox is really a statement about the non-amenability of F_2; it just happens to have a somewhat counterintuitive consequence because F_2 embeds in SO(3).
You could think of the axiom of choice as reaching into the set and blindly picking out some element. It doesn’t matter which element you choose, or even if there’s a reasonable way of computing or specifying which element you picked.
Here’s a puzzle that’s my favorite somewhat counterintuitive consequence of the axiom of choice: A (countably) infinite set of prisoners labelled 1, 2, 3, … are given a chance for release. For no particular reason, the warden is about to march them out into the courtyard, put them into a line starting with Prisoners 1, 2, 3, etc., and put a white or black hat on each one. Each prisoner can see the hats of the prisoners in front of him (1 can see 2, 3, …; 2 can see 3, 4, …; and so on), but not his own or those of the prisoners behind him. The prisoners can come up with a strategy before entering the courtyard, but no communication whatsoever is allowed once the game starts. If a prisoner guesses his hat’s color correctly, he goes free; if not, he’s executed.
Show that the prisoners have a strategy that guarantees that all but finitely many of them will go free.
I think Chronos’s objection could just as well be raised against the obviousness of those statements: Every vector space has a basis? Ok, tell me a basis for the space of countably infinite sequences of reals. Perhaps it only seems obvious because we are so accustomed to hearing others say “every vector space has a basis”, and not because we would have been inclined to think it from scratch.
The “Well, just blindly pick one” idea has more force. So, of course, what it comes down to is that there are different notions of set theory, some which formalize the intuitions behind “Just blindly pick one” and others which formalize the intuitions behind “Well, tell me which one you chose?”.
[Also, re: Tychonoff’s theorem in particular, I was convinced by Johnstone’s book “Stone Spaces” that the right way to think about it is that the product of compact locales (a topological space formalized directly in terms of its lattice of open sets, without worrying about attaching points to them) is compact, a statement which requires no use of the axiom of choice. The axiom of choice only comes up in demonstrating that if you start with “Bourbaki spaces” (topological spaces as traditionally formalized in terms of points), the resulting locale also arises from a Bourbaki space. I thought that was a nice way of understanding the situation.]
Doesn’t it imply only that if you want to, or need to, *you’ll be able to *pick a member from each set. It’s only saying that it’s possible to make such a selection. I don’t see that ‘potential’ as being counterintuitive. ETA: I’d say that denying the possibility would be counterintuitive.