This is going to be a short op for a topic I could right a lengthy paper on. I could go on forever…
I had a conversation tonight with a friend about infinity, death, and Clive Barker. I’d just like to touch on one of the things we mentioned about infinity. Are there levels of infinity?
We all know that you can keep counting forever, infinitely forwards or backwards. You may know that there is also an infinite amount of numbers between 0 and 1, because you can keep getting smaller, past 0.000000000000000000000000000000000000000000000001 and much, much, much smaller. But…
But what about the infinite amount of numbers between 1 and 10? Is there ten times the infinity between 1 and 10 then between 0 and 1? This, my friends, is a head fucking idea. I’ve been pondering this for so long that it’s even beat the Purple Ping Pong ball joke… and I spent a loooooooooong time thinking about that. (For those who don’t know the purple ping pong ball joke, you’re lucky.)
So, now I’m sitting here, thinking about going to bed, but I know I won’t fall asleep. Damn that infinity…
I do something like that with thinking about what’s outside of our universe and outside of that and outside of that and on and on…but this infinity thing isnt doing it for me
A kid goes through kindergarten, and graduates. His father tells him he can have anything he wants for graduation. “I want a purple ping pong ball.”
Alright, thinks the father. That’s cheap, fine with me.
Kid goes through grade school, same “anything you want” story. “I want a egg carton full of purple ping pong balls.”
Father doesn’t complain or ask why.
High school, ditto. “I want a milk carton full of purple ping pong balls.” Yet again, father doesn’t mind.
College, you guessed it. “I want a gross of purple ping pong balls.” Alright…
Finally, the kid gets cancer. The father offers the same deal. “I want a truckload full or purple ping pong balls.” The father is finally curious enough to ask, “Son, why do you always want the purple ping pong balls?”
Think in terms of sets and you’ll see that it is possible for an infinite set to contain another infinite set.
Let’s start with the set of all natural numbers:
0,1,2,3,…
There are clearly an infinite number of natural numbers. Now take the set of all integers:
…,-3,-2,-1,0,1,2,3,…
Note that there are almost twice as many integers as natural numbers and that the set of all integers is a superset of the set of all natural numbers. But twice infinity is still infinity.
Now consider the set of all rational numbers. A rational number is any number that can be written as a ratio of two integers, e.g., 1/2, -4/5, 3531/124. Every integer is also a rational number, since these can be written as, e.g., 2/1, -10/5. Clearly this set defines a larger infinity than the set of all integers.
Then we come to the set of all real numbers. This encompasses all the rational numbers plus all the irrational numbers. Irrational numbers are simply numbers which are not rational, i.e., you cannot express them as a ratio of two integers. This includes pi, e, and sqrt(2).
I have a friend who hopes to prove that infinity is not indefinite, and thus does not exist. If he ever does, I’ll let you know, and you can put your mind at rest.
Yes, there are lesser and greater infinities, but not quite in the way Terminus describes.
This is really GQ territory, but since Totoro asked the question here, I’ll answer it here.
As one of Terminus’ links says, mathematicians “say that two sets have the same cardinality if and only if there [is a one-to-one correspondence] between elements of the sets.” So if Tom, Dick, and Harry are loitering on the street corner, there are three guys there, because we can put {Tom, Dick, Harry} into 1-1 correspondence with (1,2,3}.
That’s overkill when you’re talking about finite sets, of course, but finity is a good place to define your terms before zooming off into the wild blue yonder of infinity.
The integers have exactly the same cardinality as the natural numbers, since we can put the entire set of integers {…-3,-2,-1,0,1,2,3,…} into 1-1 correspondence with the naturals {1,2,3,…} by arranging them: {0,-1,1,-2,2,-3,3,…}.
Each integer has been matched up with a natural number: any nonnegative integer n has been paired with the natural number 2n+1, and any negative integer -n has been matched up with the natural 2n. So it works: we’ve matched each integer with a different natural number. Therefore they have the same cardinality.
The natural reaction to this is: “That makes no sense at all! It’s obvious that there are more integers than there are natural numbers!!” The problem is, mathematicians have found no consistent, workable definition of “how many” that encapsulates this notion. With infinities, one-to-one correspondence is really the only tool we have that allows us to consistently and systematically deal with the question of whether one infinite set has more, fewer, or the same number of elements as another. So that’s what we work with: if two sets can be put into 1-1 correspondence, they are regarded as having the same number of elements.
In a similar but slightly more complicated way, the rational numbers can be put into 1-1 correspondence with the naturals. A rational number m/n can be regarded as an ordered pair of integers (m,n) with n nonzero. Turns out we can put all the ordered pairs of integers - or the ordered triples, quadruples, or n-tuples of any length, into 1-1 correspondence with the naturals.
So all these sets have the same number of elements.
“So do all infinite sets have the same number of elements?” I hear you ask. Nope, infinity isn’t that boring. The real numbers can’t be put into 1-1 correspondence with the natural numbers. (A quick proof is here.) This brings us to a division: mathematicians call infinite sets that can be put into 1-1 correspondence with the naturals, or counting numbers, countably infinite, and sets that can’t are uncountable.
The real numbers as a whole - in fact, the real numbers in any (a,b) interval of the number line with a<b - are uncountable. The reals are big compared to the rationals.
Is the cardinality of the reals the biggest infinity? Nope - mathematicians can construct sets that are too big to be put into 1-1 correspondence with the reals. There’s an infinitude of higher infinite cardinalities beyond that, too. Infinity just keeps on going. It’s enough to make even a mathematician’s head spin, at times.
Ouch! ouch! I should have put in the disclaimer that IANAM, as was working almost entirely off the top of my head. I only put in those links at the last minute.
Anyway, Totoro, did you get what RTFirefly said? There are countable infinities and uncountable infinities. You can also define an infinite number of infinite sets. :eek:
Reading this, I keep getting that “Schoolhouse Rock” song in my mind: “Figure 8 is double four/Figure 4 is half of eight/If you skate, it would be great/If you could do a figure 8/That’s a circle that turns round upon itself…Turn it on its side and it becomes/Infinity…”
I’m going to answer this in a simplistic way. No complex math or long reading. No there are not degree’s of infinity. Infinity is not a number it is a concept. Infinity is without limit by definition. So there are an infinite number of numbers between 0 and 1. There is also an infinte number of numbers between 0 and 10. Simple.
There are degrees of infinity and there is nothing wrong with considering infinity to be a number. Two sets have the same size (cardinality) if there is a one-to-one mapping from one set onto the other. You can produce such a mapping from the integers onto the rational numbers. The set of real numbers is strictly larger than the integers, because any mapping from the integers into the set of real numbers will leave some real numbers out of the mapping. For every set you can always produce a strictly larger set by taking the power set – the set of all subsets of a given set.
It’s infinite, yes. But it’s not the same (surreal) number.
I told you I shouldn’t mention surreals. One of their neat properties is that you get infinite numbers that are quite a bit smaller than you’d expect. The logarithm of omega is a particularly neat example.
If you’re pathologically interested, like I am, pick up John Conway’s book On Numbers and Games. He does an infinitely better job of explaining them than I could.
