I give. What’s green and very far away, and what’s grey and ignored by Grothendieck?
As has been pointed out, there’s no possibility of actually doing this. There are good reasons to accept the axiom of choice, though–you can see some discussion in this very old thread. Since that thread, I’ve learned that one of my undergrad professors doesn’t like the axiom of choice (I believe because of his work in orthonormal polynomials, but it’s been long enough that I’m not certain about that), but he’s definitely in the minority.
By Googling, I discovered that it’s the lime at infinity that’s green and very far away. I haven’t been able to figure out what’s gray and ignored by Grothendieck. I haven’t been able to guess at it either. I’ve tried looking at some articles about Grothendieck, but none of them talk about something he ignored. I believe in jokes of this sort when you say that something is gray, you’re usually looking for a pun on “elephant”, just as when you say that something is purple, you’re usually looking for a pun on “grape”, but I can’t think of any mathematical term that puns on “elephant”.
Google will get you the first one, but Mathochist is going to have to enlighten us on the second.
The irrelephant ideal.
Incidentally for those who don’t want to Google, the other answer is the lime at infinity.
Ah, but any finite number of cuts can only give you a measurable set, and as has been pointed out the pieces are generally not measurable.
In fact, at least one of the pieces (two, if I recall) is a Cantor dust. That is, it’s topologically equivalent to the Cantor set.
Well, again the rule of thumb is “AC (or equivalent) is really good in algebra and really weird in analysis”. It’s also far from clear why it should hold.
Still, I like my solution of having different topoi obeying different logics, and letting the mathematician in question choose which to use and develop the mathematics of. It’s analogous to the situation in geometry, where the parallel postulate can be accepted or rejected and different valid geometries spring from each choice. Then we just leave it to the physicists and philosophers which option obtains in the “real world”, and get back to doing some math.
Of course… I had forgotten that sequences were allowed to be infinite. Is the bound required to be absolute? Because if so, I’m having a hard time thinking of any candidate spaces at all.
And in case anyone is wondering, Abelian grapes are purple and commute, and Zorn’s Lemmon is yellow and equivalent to the A. of C. I knew those two, had to Google for the lime at infinity (which is incidentally very rare, even on Google), and had no clue about the irrelephant ideal.
Sounds like the punchline to the old joke about having both a wife and a mistress ;).
From the thread I linked to:
Anyone know this one?
The well-ordering principal.
Can’t believe it took me five years to get that. :smack:
“Spaces”. Are you thinking topological spaces? That’s far too much structure.
One place it comes up a lot is in algebraic geometry. Consider the set of ideals of a Noetherian commutative unital ring R containing a fixed ideal I, partially-ordered by inclusion. The Noetherian condition is equivalent to stating that any increasing set of ideals terminates, and obviously the top element is an upper bound. Thus (by Zorn’s Lemma) there exists at least one maximal ideal P of R containing I. There are all sorts of similar applications in that field.
Of course. We’ve got certain obligations to physics, and we screw around with philosophy for fun.
The Well-Ordering Principal.
No, I’m thinking “Big ol’ set that you do a bunch of interesting stuff inside of”. Forgive me if I’m not more specific, but I’m a physicist, not a mathematician.
And the punchline I was thinking of was “It’s better to have both a wife and a mistress, because that way, your wife will think you’re with your mistress, and your mistress will think you’re with your wife, and you can stay up at the office and do math.”.
I know. I was drawing an analogy and running with it. When you distract the physicists with philosophy and the philosophers with physics you can be left alone to do some math.
At work today, I asked another mathematician who knows a little more about the sort of thing that Grothendieck does what the answer to the riddle could be. He said that Grothendieck did geometry in a very abstract fashion. It’s almost as though you could forget about everything you learned by reading Euclid. So I came up with the following:
Q.: What’s gray and ignored by Grothendieck?
A.: Euclid’s Elephants.
I also came up with the following:
Q.: What wades in the river and vanishes only on a single line?
A.: Riemann’s hippotamus.
Well, Grothendieck worked in algebraic geometry, which is (or at least was before Grothendieck) more about using geometric techniques to solve systems of polynomial equations. Euclid never really came into it that much.
Considering the number of people who can lick their own elbows, as evidenced by the all the photos Brainiac got when they made the statement that no one can, that theorem wouldn’t have to be very complex. Or counter intuitive.