This is the set of integers: {...,-3, -2, -1, 0, 1, 2, 3, ...}
Consider the duck as a 3-dimensional solid whose surface has uncountably many points. Therefore, while the integers can be listed (yes an infinite list but an organized list), the duck-points cannot be placed in an organized list.
Both sets (the integers and the duck points) are infinite sets. But because the integers can be placed in an organized list but the duck points cannot, then we say that the set of integers has size aleph-null but the set of duck points has size c. (Aleph-null and c are different sizes of infinity and are called cardinal numbers. You math people can read a book on set-theory and it talks all about it.)
It is a fundamental principle of set-theory that aleph-null is smaller than c.
The bottom line: The set of integers weighs less than a duck, and is therefore a witch. QED
Please forgive my brief moment of insanity.
But if the duck points are listed in any other organizational pattern other than as a duck, don't they cease representing a duck and become something else? Like a tree toad a toaster? I guess I'm saying the organizational pattern of the duck points are what makes a duck a duck.
ReplyDeleteAnd, since ducks float, if the weight of the set of integers is less than or equal to that of the duck, wouldn't it still be a witch?
Of course, I could be wrong on all counts. But don't tell me if I am 'cause it would damage my self of steam.
But if you use a principal component analysis on the duck, you will find it reduced to a finite set of watermelon-like objects. This is shown as figure 37(b) on page 23 in the section comparing NAPCA to PCA via RX methods.
ReplyDeleteErgo, give me a Ph.D. and a job!
Didn't Cantor use a diagonal argument to show that the set of real numbers (the duck) is necessarily denser than the set of integers? Everyone knows that dense objects don't float. So neither do ducks.
ReplyDeleteI understood c. 50% of that, but found it very funny nonetheless. Bravo!
ReplyDeleteQuestion from student who is lost, in over her head, and trying to sidetrack discussion: what happens when the duck in the picture mistakes the integers in the list below for insects, and starts eating them?
@ Contingent Cassandra:
ReplyDeleteWe would have something similar to the "infinite hotel" scenario. In the infinite hotel, if all of the rooms are occupied and somebody needs a room, then you can have each guest move to the next room and then a room is freed up.
There are still infinitely many rooms occupied, so infinity = infinity + 1.
If the duck started eating some of the integers, we would still have the same amount of integers left over as we had to begin with.
The bottom line: What happens if the duck starts eating the integers? Absolutely nothing, in terms of size.
The duck would be happy as he would have discovered an infinite food supply!
But a point has neither length nor width, and thus is cannot have mass, so it can't weigh anything. If a duck can be described as a set of points, then no matter how many points it's made up of, it will never weigh anything and can neither float nor sink. If it doesn't weigh anything, then nothing can weigh less than it, including itself. So the set of integers cannot weigh less than a duck and is not a witch.
ReplyDeleteFrontDoor, perhaps the point is moving at or near the speed of light. Then it could have a relativistic mass. Of course, if a witch traveled close to the speed of light, she would probably end up hitting a tree because it would be hard to steer her broom.
ReplyDeleteEMH, it seems that the size of the integer set depends on whether it can be organized (aleph-null) or not (c). Does the size of an integer set depend on the number of integers in the set and whether it is organized?
If you can talk about different sizes of infinity with a straight face, we need to get stoned and hang out. That would be a blast.
But, Front Door, if the duck *eats* the integers, then the weight of the duck would be duck + integers, which is > than integers alone.
ReplyDeleteBurn her anyway!
ReplyDeleteBen, the fact that they can be listed "completely" makes them what we call countable. There is only one kind of countable infinity. Any other organized list of the integers will be in one to one correspondence with the first list. So they are the same size.
ReplyDeleteThe ability to place two sets in one to one correspondence (i.e. complete monogamy of both sets) is how we decide if two sets have the same size.
It is interesting to note that the set of positive integers, the set of all odds/evens, and the rational numbers all are the same size as the the set of all integers.
This is the funniest thing I've read in weeks.
ReplyDeleteCan I use this in my 7:30pm calculus class tonight?
ReplyDeleteThank you, EMH, this is the high point of the week! And it proves that math is useful. Ducky!
ReplyDeleteI second what Suzy said. Math is useful and interesting. Apparently, it is also monogamous, which may be why it's so hard for many people.
ReplyDeleteI dunno. Some people think that being "so hard" is a plus, especially for your monogamous partner.
ReplyDeleteQED = Quack-Emitting Duck
ReplyDelete@ Suzy,
ReplyDeleteBy all means do so!
@ Ben,
Nothing brings me back to graduate school quicker than having some scoobie snacks with a colleague!
I must take issue with your modeling of a duck as a three-dimensional solid with an uncountable boundary. To me, a duck is a finite collection of meat and feather molecules. If anything, a duck should weigh less than the set of integers.
ReplyDelete(Edited for clarity.)
I think EMH has found his genre: the quack-proof, which can be defined either as a proof involving a duck, or a proof that, on close inspection, does not quite hold water (and may or may not float, be a witch, and/or be liable to burning).
ReplyDelete