| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 743.1 | | CLT::GILBERT | Builder | Wed Jul 29 1987 16:01 | 5 |
| That theorem can be derived from the following conjecture:
For any prime p, there another prime q with p^3 < q < (p+1)^3.
Is this conjecture true?
|
| 743.2 | Not Exactly A Theorem, But ... | COMET::ROBERTS | Peace .XOR. Freedom ? | Thu Jul 30 1987 16:11 | 5 |
| How about a 3-dimensional object that has a finite volume but an
infinite surface area; i.e., you can fill it with paint but can't
paint it? I wish I could remember what this object looks like or
is called!
|
| 743.3 | Use a BIG brush | AKQJ10::YARBROUGH | Why is computing so labor intensive? | Fri Jul 31 1987 12:33 | 8 |
| > How about a 3-dimensional object that has a finite volume but an
> infinite surface area; i.e., you can fill it with paint but can't
> paint it? I wish I could remember what this object looks like or
> is called!
There are several of these. The simplest is the 'bucket' formed by rotating
the curve y=1/x about the x-axis. I always heard it called the "pathological
paint-bucket".
|
| 743.4 | Infinite surface/ finite volume | RDVAX::PERRONE | | Fri Jul 31 1987 12:45 | 4 |
| re: 743.3
You have to bound the "bucket" away from zero or it will have infinite
volume. Use the interval [a,infinity) a>0
|
| 743.5 | | CLT::GILBERT | Builder | Fri Jul 31 1987 14:13 | 1 |
| Another such object is the Sierpinski sponge.
|
| 743.6 | Even integers ... | ESASE::HEGARTY | | Fri Oct 09 1987 09:24 | 12 |
| Even though it isn't strictly a theorem, I feel it is definitely counter
intuitive to be able to prove there are as many even integers as there
are integers!
Proof:
======
List the integers ... -3 -2 -1 0 1 2 3 ...
Now multiply each by two to give,
... -6 -4 -2 0 2 4 6 ...
Thus, corresponding to each integer there is an even integer.
|
| 743.7 | | ULTRA::ELLIS | David Ellis | Mon Oct 12 1987 13:33 | 12 |
| Re: .6:
The property mentioned is a one-to-one correspondence between the set of
all integers and the set of even integers. This is a special case of the
(counter-intuitive?) theorem that any infinite set can be placed in one-to-one
correspondence with a proper subset. The theorem is clearly false for
finite sets.
In fact, one can even _define_ an infinite set as a set for which there
exists a one-to-one correspondence with a proper subset.
David Ellis -- Secure Systems Group -- LTN2-2/C08 -- DTN 226-6784
|
| 743.8 | The dreaded AC strikes again! | ZFC::DERAMO | Daniel V. D'Eramo | Mon Oct 12 1987 21:07 | 19 |
| .7>> any infinite set can be placed in one-to-one
.7>> correspondence with a proper subset.
The famous "axiom of choice" (a weak form known as the
countable axiom of choice is sufficient) is needed to prove
this.
.7>> In fact, one can even _define_ an infinite set as a set for which there
.7>> exists a one-to-one correspondence with a proper subset.
The countable axiom of choice is also needed to prove that
this definition is equivalent to the "usual" definition of
"infinite set" [roughly, an infinite set is one that is not
finite, and a finite set is one that is either empty or "has
N elements" for some positive integer N -- i.e., can be put
into a one-to-one correspondence with the set of positive
integers less than or equal to N for some positive integer N]
Dan
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| 743.9 | Godel's Incompleteness Theorem | PRCS85::EDDIELEUNG | NO Artificial Intelligence Added | Fri Jun 10 1988 10:02 | 16 |
| How about Godel's Incompleteness Theorem ? I was shocked when I
first learnt of it. Even now, my "Mind" still can't follow my "Brain".
Put in English, the theorem says that for any consistent logical
system rich enough to describe elementary number theory, there exist
a well-formed (meaningful) sentence within that system such that
it can neither be proved nor disproved using only the machineries
provided by that system. However, intutively most people would
say that the sentence is *true*.
The mind and the brain fights again.
Eddie Leung.
Software Services.
|
| 743.10 | | VINO::XIA | In my beginning is my end. | Mon May 27 1991 07:56 | 43 |
| It is kind of late, but any way here goes:
re .9,
The statement is not just true "intuitively", but the validity can be
reached through meta-reasoning. In other words, the statement is
absolutely true. The argument goes like this. Godel first formally
constructed a sentence H which is "The validity of the formal statement
H can not be demonstrated within the formal system". Now here comes
the perverse part of the thing. :-) Godel showed that that very H
sentence is unprovable within the system (the "why" is left to the readers
as an excercise. :-) Seriously, it isn't hard at all)! But that very
proof shows that statement to be true because that is what statement H
says!
...
My candidate for the counter-intuitive things is follow.
Take a unit square, and you can inscribe four equal sized circles tangent
to the sides and to each other. Then you can inscribe another circle
in the middle that is tangent to all other four circles.
Similarly, you can do the same with a unit cube, i.e. you can inscribe four
equal sized spheres tangent to the surface of the cube and to each
other. Again you can inscribe another sphere in the middle that is
tanget to all other four spheres.
And for a hyper cube you can do the same thing, and on and on...
So for a general unit n-cube you can inscribe four equal sized
n-spheres tangent to the surface of the cube and to each other.
Furthermore, you can construct another n-sphere in the middle that is
tangent to all the other four spheres.
Now here is a surprise. As you increase the dimension, the radius of
the sphere in the middle increases and soon becomes larger than 1 (!)
which is the length of the side of the unit cube. In other words, the
n-sphere in the middle "pops" out of the cube as n becomes large
enough. In fact, you can prove that the radius of the middle circle
goes to infinity as n goes to infinity!
Eugene
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