This question is directed at anyone who is looking for the “answer”.
Why? I don’t mean to bring religion into this because I believe religion doesn’t give anyone the answer it just makes mortals content with the knowledge that at least someone somewhere has it. I’m talking about US knowing. Hasn’t everyone see enough of life in general to know that not only is it unfair, unruly, and dangerous, but ultimately absurd. Here’s a thought. OO is infinity, or a number past human comprehension. If OO drifts in a realm to large to count what’s OO - 1 ???
Just read Catch-22, had to see what this board has to say.
I really hate thread which give you no clue as to what the question is. If not for the OP’s inclusion of that final line about Catch-22, and me happening to know that Catch-22 was written by Joseph Heller, I’d be totally mystified by what Occam is asking. As it is, I don’t know who Douglas is, and I still don’t understand the question.
I’m not a mathematician, but I was knee-deep in it in my EE studies, that’s for sure.
I too don’t completely understand the question.
oo doesn’t ‘drift’ into numbers too large to count, oo is oo. Now a number, say ‘x’, can drift into ‘numbers too high to count’, and if I understand your question correctly, I would state that as ‘x->oo’. And then you ask what if ‘that number’ that is drifting into numbers too high too count, what is that number - 1.
Using my paraphrasing, if ‘x->oo’ what is ‘x-1’? (x-1)->oo-1, so (x-1)->oo.
Forgive me if I’ve gone on a mathematical spiel that you intended to be a philosophical discussion, but it seems to me to be a poorly phrased question that really isn’t that much of a ‘great debate’, it can be addressed quite easily in mathematics.
Now, if you were really trying to ask ‘what does infinity really mean’, I can’t help you there. My limited human brain isn’t capable of fully comprehending the concept, that’s why I use the symbol oo.
Now, this is the part where you have to tell your brain to scram, and instead let your heart do the math. Yes, that’s what I said. This kind of math is for the heart, not the brain. In your heart, you know that the two-inch line has twice as many points as the one-inch line has.
Your brain is yelling back, “That’s impossible! They’re both infinity!” Tell your brain to shut up and mind its own business. Your heart understands that there can be different kinds of infinity, and some are bigger than others.
Like for instance, OO-1 is smaller than OO. Not in any practical way of course, but the heart is not bothered by what’s practical or not.
On the other hand, Georg Cantor “proved” that the infinite number of rational numbers is “less” than the infinite number of real numbers, and that in turn is “less” than the infinite number of curves. I boggled down when I first started trying to understand the theory of transfinite numbers.
Yes, this is definitely true as polycarp points out in his post. But for your example, it is, unfortunately, not true. Both infinities are the same size. I like the hotel analogy. For infinity and infinity+1 it goes like this. You have a hotel with an infinite number of rooms numbered from 1 to infinity. All rooms are occupied by guests, one to a room. Suddenly, a new guest shows up and needs a room in the infinite hotel. Is there room for him?
The answer is that you simply move every one up one room (that is, 1 goes to 2, 2 goes to 3, etc.) and put the new guy in room 1. Since we are dealing with infinity, there is no point at which this breaks down, so everyone gets a room. This proves that infinity and infinity+1 are the same size.
However, say an infinite number of guests shows up (it’s a busy day at the hotel). To accomodate them all we do is to order the new guests from 1 to infinity. Now we take the existing guests and move them into the even-numbered rooms (that is, move 1 to 2, 2 to 4, 3 to 6, etc.). Then we take the new guests and move them into the odd rooms (that is, move guest1 to 1, guest2 to 3, guest3 to 5, etc.). Now everyone has a room, and therefore infinity+infinity=infinity.
Of course, this analogy only applies to countable sets like the integers or the rational numbers, but a similar argument can be constructed for the real numbers.
your humble TubaDiva
Only giggling a little bit . . . promise.
PS While we’re on similar subjects, anybody read the new Richard Feynman? He’s more prolific dead than when he was alive! HEY! He’s managed to conquer space and time . . . if anybody could, I guess he’d be the man. . .
43rd Law of Computing:Anything that can go wr {segmentation fault}
If you are correct, any non-existent numbers above 42 are merely the product of the diseased imaginings of a six-legged Tanterbok suffering from the consumption of one too many (no such thing!) PanGalactic Gargle Blasters, and all reality will disappear 30 seconds after Marvin comes in and gives said Tanterbok the boot out the airlock.
<FONT COLOR=“GREEN”>ExTank</FONT> <FONT COLOR=“RED”>“WE APOLOGIZE FOR THE INCONVENIENCE.”</FONT>
Polycarp’s Law of Web Posting is that the flames shall not exceed the puns, and the examples of true insight are always less than the sum of the above. Apply that reasoning to this thread.
I bet an infinite number of guests would be quite pissed off if they all had to move their rooms because more guests checked in. Imagine the lines for the elevator and stairs.
Ah, but Ursula, a hotel with an infinite number of rooms would have an infinite number of elevators! Unfortunately, it would also have an infinite number of floors, and all the infinite elevators would be at other floors than yours (probably an infinite distance away)…
I realized from Polycarp’s post that my experience with mathematics in my studies was fairly restricted to the practical and not the theoretical side of math. Taking a clear opportunity to improve myself, I did some (admittedly superficial) research on Georg Cantor’s transfinite theory.
From my understanding, this theory deals with the cardinality of sets, and can define relationship between the sizes of the sets. From what I read, it wouldn’t apply to the case of ‘what is oo-1’. For one thing, subtraction isn’t a definable operation in the cardinal number arithmetic Cantor provided. Even so, his theory stated that aleph_0 (the set of all positive integers) + n (some finite cardinal number) still equals aleph_0. Differences in size between the sets doesn’t come into play until you take some cardinal number n (>1) to the power of aleph_0 (or aleph_1, etc).
Keeves said:
I’m assuming, Keeves, that you make this statement because: “x-1 < x”, which is true for all finite numbers, but I don’t think is true of oo.
Let’s assume x > 0. (I think the fact that oo > 0 may be something that everyone here can actually agree on :)) This lets us divide by x. Dividing the above by x, we get:
1 - 1/x < 1 , which is also true for all finite numbers greater than 0. But taking the limit as x->oo, we get:
lim(x->oo) 1 - 1/oo < 1
or 1 - 0 < 1. I think another thing that everyone here would agree on is that 1 < 1 is a false statement. I’m not sure if my taking the limit as x->oo changes the rules, but thats the only way to do arithmetic with oo that my understanding of mathematics allows.
Any actual mathematicians out there? The deeper theoretical stuff is one of the reasons why my high school enjoyment of math turned into studying an area that concerns itself with the practical application, instead of purely studying math.
Here’s another puzzle for the math inclined.
The surface area of a one unit cube is 6
The area of a one unit cube is 1
S>V
The surface area of a 10 side unit cube is 600
The volume of a 10 side unit cube is 1000
S<V
The reason is of course that S = 6LW
whereas V = (L)/\3
OK…a ‘unit’ is right now an arbitrary measurement. The laws of math are beholden now to what we describe as a ‘unit’. Fine.
How does this change reality then when we make a unit 1000 light years. Or we make a unit 1 cm? Do things have more surface area to volume when we change the units?
One of the foremost theoretical astrophysicists/cosmologists of the 20th Century (IIRC, Eddington, and I welcome corrections) spent the last few years of his life searching for the significance of the number 137. He had not gone wacko (or so people think) but had simply noted about a dozen occasions in nature in which the “raw” number 137 occurred and was looking for what possible connection there could be between them. I don’t remember most of his list; IIRC, one was the ratio of the strong nuclear force to the electromagnetic force.
Consider the number of silly places Planck’s constant crops up, and you may see his point.
