If the universe is "expanding", how can it be infinite?

I think most people would see the clear fact and agree that the number of irrational numbers larger than zero and less than one equals the number of irrational numbers larger than one and less than two and so on.

Quite clearly there are more irrational numbers between zero and two than between zero and one. In fact, there are twice as many.

I am not sure why you are arguing against this.

I am sure that your explanation is just silly

Well, not exactly. Clearly, the smaller of the two sets is contained within the other. That said, since both sets are unaccountably infinite, they have the same order and are "the same size."

The problem comes from the fact that you can't simply add together infinite sets and change their size. A set of order 1 combined with a set of order 1 yields a set of order 2; an unaccountably infinite set combined with any other set is not subject to the same rules of arithmetic: its size does NOT change.

You could say I'm arguing against it because it's not true. It really isn't. If you think "common sense" can reason you the correct answer about this you're wrong. In no sense are there twice as many irrationals between zero and two than irrationals between zero and one.

I'll try explaining it another way. Consider pairing up the irrationals between zero and one with the irrationals between zero and two. If there really are more irrationals between zero and two then we would have many unpaired irrationals. But here's a way to pair up every last irrational between zero and two with a unique irrational between zero and one: divide it by two. An irrational divided by two is still irrational and it will be between zero and one. So for every irrational between zero and two there is a unique irrational between zero and one. Therefore there are no more irrationals between zero and two than there are between zero and one.

I hope that helps!

it doesn't work like that

i have tried a lot to understand this & i'm still long way from there

what cantor/etc are saying is that integers are lowest set of infinity

rational numbers ( no matter between 0 - 1 or 0 - 100 ) are next level

irrational numbers ( no matter between 0 - 1 or 0 - 100 ) are even higher

it's nothing to do with the interval considered, but "matching" quality of the different levels

i e for every "rational" e g

3.33

we can have irrational "higher" set of

3.33* ( from 10/3 )

3.333* ( from 10/3 with further d.p at "infinity" ?! )

3.333* ( " " )

3.3333* ( " " )

3.33333*...

the "rational" number 3.33 doesn't have an "equal" number of "matching" numbers in the set of "related" "irrationals"

the latter set is of a higher level of infinity

anyhows, if anyone knows of a recent book in "pop science" category dealing with this, please post it, as i must buy it !!!

Is it just me, or does this thread come up on the boards at least once a month?

agree wrote:

Is it just me, or does this thread come up on the boards at least once a month?

You're not the only one who comes on the boards at least once a month. So no, it's not just you.

If there's one thing I've learned from "Through the Wormhole with Morgan Freeman", it's that the Universe isn't infinite.

surprise macaroni surprise wrote:

doesn't seem to make sense

What's one degree north of the north pole?

men in black

Base the theory on a synthetic system, not analytic. Use mass, not numbers. Use the ideas of finding a way to balance space using weight instead of balancing with numbers.

Not quite. There exists a bijection between the natural numbers and the rationals, so they are of the same cardinality. If you want to be technical, you have to show there exist bijections:

N --> N x N --> N x Z --> Q,

where N are natural numbers, Z are the integers, and Q are the rationals. Since the composition of any number of bijections is a bijection, it follows that there exists a bijection N --> Q, and hence they are "equally" countably infinite.

Todd: The issue is what precisely do you mean when you're saying there are more numbers. Math is all about coming up with and handling very precise definitions.

It looks like you're saying there are more irrational numbers in [0,100] than in [0,1] in the sense that every irrational number in the latter set is a member of the former. This is true, and you can use this as a way of saying if something is more than something else, but it turns out "inclusion" as it is called is not really what we mean by size. For instance the set {1,2,3} is the same size as the set {4,5,6}, but neither is contained in the other, so inclusion doesn't say anything about their relative size. By our normal notion of size of sets they should each have the size 3.

It turns out that a useful notion of size is constructing "bijections", that is, finding a way to pair each element in one set with exactly one with the other set. The thing with infinite sets is even if you have a bigger set by inclusion, they can be the same size according to bijections. For instance, all integers and the even integers have the same size using bijections (pair the two sets according to the map that sends x to 2*x).

You guys are unbelievable. Actually arguing there are as many irrational numbers between zero and one as there are between zero and one hundred and offering goofy examples.

point of fact wrote:

lol, discussing pointless sh*t like this is exactly why you aren't getting laid.

Yes, that's a nice view on life! It's all about copulation, everything else is pointless and an utter waste of time.

Surely you jest.

When it comes to infinite numbers, "counting" is quite different.

In fact, the number of irrationals between 0 and 1, is the same as the number of irrationals between 1 and 2, and in fact, it is the same as the number of irrationals between 0 and 100, or between -1,000,000 and 1,000,000. The number of irrationals in any finite open interval is the same infinity.

People above are correct, to "count" infinities we match up elements from each of the sets. This means that you can take every irrational in (0,1) and match it up with every irrational in (0,100) in such a way that every irrational in (0,100) gets matched to, and no two irrationals in (0,1) are matched with the same irrational in (0,100).

Here's a famous example to see why counting infinities are not so easy.

Consider an infinite hotel, with an infinite number of rooms lined up down an infinitely long hallway. No, suppose that every room is occupied. A traveler arrives, and asks for a room, and the front clerk explains that there are no vacancies. However the traveler realizes that this is an infinite hotel, and asks every guest to move one room down. So the guest in room 1, moves to 2, the guest in room 2 moves to 3, and so on. This opens up room 1, and the new traveler arrives.

Now, the next night, an infinite bus arrives, carrying a countably infinite amount of passengers. The hotel clerk tells the driver that he has no vacancies, and surely he cannot accommodate another infinity of guests. However, the bus driver realizes that there is a way to free up rooms. "Surely, there must be an infinite number of odd number rooms, and an infinite number of even number rooms" says the bus driver. "Indeed," replied the clerk. "Well, if everyone in room n moves to room 2n+1, then all of your even rooms will be empty, and my passengers can stay in the even rooms.

Clearly, even a hotel with 1,000,000 rooms cannot be run in the same fashion as an infinite hotel.

jibberish

surprise macaroni surprise wrote:

doesn't seem to make sense

We've only been to the moon and have never set foot on another planet. Nor have we discovered any other form of life, outside of earth. That is comparable to only visiting the town next door (once), yet expecting to know the world.

Man is still in his infancy. Infants don't know much.

Adding "in fact" to a sentence just before you make some ridiculous claim does not make it so.

And what the hell was that story supposed to add?