The End of Infinity.

one way ive thought about infinity is as the edge of the universe. if the universe isnt infinite and is expanding then what is the edge of the universe? whats beyond it? thats when my brain peters out.

I actually do believe a bar with one end could be infinite. Hear me out.

There are an infinite quantity of real numbers; similarly, in an infinite universe, there are no bounds to three-dimensional space.

Given that there are an infinite quantity of real numbers, and that any fraction of an infinite quantity is, by definition, infinite, then there are also an infinite quantity of positive real numbers; infinity can be terminated at one end, but not at both.

That's just my thought, anyway; feel free to contest it if you like.

Amantine is correct about the bar:

You would be holding the bar at or near the beginning of an infinity.

As to the cardinality of the number line:
All positive-integers are countable (aka aleph 0 or whatever notation he is using)

card(pos. and neg. integers) = card(pos. integers)
card(all rationals, +and-) = card(pos.integers)
card(irrationals) not countable (aka aleph1)

the union of two countable sets is still countable.. blah blah

but the 2^aleph0 thing i seriously doubt, unless i am wrong in my assumption that aleph1 is the uncountablt infinity we discuss in our transfinite courses here at ucb.

The reason is that the cartesian space has the same cardinality as the number line. And if you constrain your cartesian space to include only rational numbers, then even an infinite dimensional space still only contains a countable infinity of points.

I agree with Seapeople in that this is a very confusing topic. I do not claim to be an expert in classical or quantum physics, but from what I have read the concept of infinity could be somewhat easily explained by quantum phenomenon (at least as much as it can be in our dimension of existance). I could be completely off on this, but here is what I think. According to quantum phenomenon (if I understand it correctly) things are in a sort of flux until actually percieved or measured. In your example you were holding a bar that went on forever. If it went on forever you would not be able to see the full length, so until you viewed, measured, or somehow maybe on another level of existence could sense the entire pole, it actually would not be infinate at all or at least you would never be able to prove it because every length of pole you viewed, there would be more. I think that in our level of existance we are incapable of percieving infinity because we can only understand the limitations of our dimension.

Hmmm... really didn't answer anything, but was fun thinking about it for a while.

Thanks for the topic.
Kata

Hmmm.... I have a problem with the satement of "holding the bar at the beginning of infinity" just because if the bar was infinate, then there was not ever a beginning of the pole because the pole spans forever and always has.

the only place infinite can exist is in the mind. sure there are numbers that go on forever but u can only "see" it in the mind

Originally posted by Kata
Hmmm.... I have a problem with the satement of "holding the bar at the beginning of infinity" just because if the bar was infinate, then there was not ever a beginning of the pole because the pole spans forever and always has.

I have a problem with the statement "just because if the bar was infinite, then there was not ever a beginning"

Something being infinite does not mean it has no beginning, it just means it has no end.

and @silQ: i do not agree that "infinite" exists only in the mind. i think to label our physical universe that way is rather childish, especially since aspects of our reality can be demonstrated to be infinite. Indeed, i can only demonstrate this within your mind or mine, but the contrapositive belief that everything in the universe is finite can also only be demonstrated to someone within his or her mind.

[Edited on 6-4-2004 by Cascadego]

Ok, you guys are on the right track, but you have to understand that there are orders of infinity. This is how "set theory" became so popular. I'll give you a link so that you can understand the orders:

Cantor's Set Theory

A general history of Infinity

To sum up some other problems, if you have a steel bar, if it has one infinite component (infinite length), the other dimensions are also infinitely large when associated with the infinite dimension. What does that mean? The bar has infinite area and volume. Even with finite boundaries, the infinite boundary (length) overshadows the other boundaries when computing the whole. So, as long as the other dimensions exist, a steel bar of a 1 inch diameter is equal to a steel bar with a 2 inch diameter... or is it? I guess it all depends on how you define and classify infinity.

One major problem with the idea of infinity is that nothing in reality has an infinite property. So, you will never have a bar that is infinitely long, because there is not an infinite amount of material. If there were, you'd assume that all of life would be a solid? Again, it depends on how you define infinity.

[Edited on 6-4-2004 by Protector]

a line for example

is the same as
*----------------------------->

(where * means a beginning)

I think the term infinity would be usefull when describing cycles, because cycles, if given 'infinite time' can exist. Obviously it would be impossible to imagine infinity without some sort of mathematics, because you can never actually approach infinity, (e.g a point on one of the above lines) because you would still have infinity to go, but then you aren't really going anywhere because there is no end.

But would infinity need time? Because if it was a true infinity, it would be instantly created, but how can you instantly create something that goes on forever? It would seem impossible, but something that takes time to be created could be an infinity, a potential infinity, because with an inifinite amount of time it wont ever reach it but it will go on forever which is a concept of infinity. But then where does one get infinite time? From a cycle of time!


Hope this got some people thinking again, or confused some.

Originally posted by Cascadego
Amantine is correct about the bar:

You would be holding the bar at or near the beginning of an infinity.

As to the cardinality of the number line:
All positive-integers are countable (aka aleph 0 or whatever notation he is using)

card(pos. and neg. integers) = card(pos. integers)
card(all rationals, +and-) = card(pos.integers)
card(irrationals) not countable (aka aleph1)

the union of two countable sets is still countable.. blah blah

but the 2^aleph0 thing i seriously doubt, unless i am wrong in my assumption that aleph1 is the uncountablt infinity we discuss in our transfinite courses here at ucb.

The reason is that the cartesian space has the same cardinality as the number line. And if you constrain your cartesian space to include only rational numbers, then even an infinite dimensional space still only contains a countable infinity of points.

Well, 2^aleph0 = aleph1 if you choose to support the Continuum hypothesis: that aleph1 = c. You can choose yourself, because both supporting it or saying it isn't true work in ZF set theory. The generalized continuum hypothesis is 2^aleph(n) = aleph(n+1).

To solve this problem, I'll just use c instead of aleph1 and you can choose yourself if you want c = aleph1. C is the continuum and it cardinality of the set of all real numbers. It is also true that:

aleph0^aleph0 = c

Well all I know is if it has an end, it is finite. Therefore the end of infinity can not exist. And if it does not exist, how can it be there to begin with. Great thread, excellent site referals.
I love oxymorons, above all other morons :lo

That's all folksl:

Tut Tut

@tut tut... whut?

How exactly do you mean that something without end not exist?

Certainly it can, observe:

Lets count numbers, starting anywhere we want.

If 0:
0...1...2...3...4...5....6......... forever.
Whoa #, theres no damn end. But damn.. those numbers all still exist, and so does the generating sequence, and so does the set containing them, and so does the result of adding them all up, and so does the result of adding the first and subtracting the second, and so does the result of multiplying them all together, and so does the result of dividing them all by each other, and so does the result of combining this set with another.

If -23049:
-23049...-23048...-23047................forever.
Whoa # there it is again.. no end, but damn.. theres definetly a beginning... and i know those numbers exist

______
Seems to me there are as many different infinite things with beginnings and no ends as there are numbers... and that is of course to say there are an infinite number of infinite things with beginnings and no ends.



[Edited on 7-4-2004 by Cascadego]

[Edited on 7-4-2004 by Cascadego]

Originally posted by amantine

aleph0^aleph0 = c

Hmm... you are right according to the outside world, but the world in my head isnt quite so steady, perhaps you will help an undergrad better intuite why.

Basically aleph0^aleph0 implies a countable union of countable sets, no? I mean, there is a clear function associating the two sets, why then does this diverge into uncountablitliy?

Or is aleph0^aleph0 = aleph1/c simlply an axiom of set theory i've neglected?

maybe he is referring to physical things, but will we ever know without refferng to mathematics? because i doubt we can actually observe and identify an infinity without referring to mathematics(because you cant measure an infinity)

Originally posted by quiksilver
maybe he is referring to physical things, but will we ever know without refferng to mathematics? because i doubt we can actually observe and identify an infinity without referring to mathematics(because you cant measure an infinity)

Wouldn't infinity be the measurement of infinity?

Originally posted by TheBandit795
(...)
If you traveled a straight line in those dimensions, you will eventually reach back to the exact place you first were.
(...)

Even if you consider only 3 dimensions...

Makes me think of this example that I like a lot:

Suppose a plane can fly 40000 km without needing refueling. Let the plane fly at constant altitude and straight ahead. Everyone will agree that the plane is flying a straight line. But if that plane carries on that straight path for 40000 km (or about that), it will eventually end up where it was already! From an observer on the moon for example, the plane didn't fly actually straight but did fly a big loop.

The gravity around earth is responsible for that, obvisouly, and thus if the plane was flying around a planet with no gravity at all (impossible?) then it would need to correct its fly path regularly to go around it. So gravity bends indeed space, and that plane, though flying a straight line, will loop.

Ain't that a bit of infinity?

Odd thing about infinity...........

It has to have a starting point, otherwise it would just be a circle.

Let's use the metal pole as an example.

If you had a metal pole, that just kept going, and going, and going, it would have to have a starting point, otherwise it would just be a really big loop.

Cause if that infinite pole wans't a loop, then it would have to have a starting point. And if it does have a starting point, then if you were traveling along that pole going backwards, that infinity, would end..............at the staring point.

So no, infinity doesnt exist.

One more question:

10 / 3 = 3.333333333.... going on for infinity.

How can that be? If there are 3's for infinity wouldn't they all add up to be infinity and not 10?

Originally posted by Faceless
One more question:

10 / 3 = 3.333333333.... going on for infinity.

How can that be? If there are 3's for infinity wouldn't they all add up to be infinity and not 10?

Not really, since the further away from the decimal place you go, the smaller the value you are adding becomes. So as you approach infinity, you are adding an infinitely small number to the total which results in no change to the value.

You could say the same thing about walking across a room to a wall. If each time you move, you gained half the distance to the wall. How would you ever reach the wall? Can you half the distance an infinite amount of times. No, actually there is a limit to this process. In math you would say...(The limit as ten divided by 3 approaches infinity) Lim x/3 --> oo = in this case 3.333333....

You can really have fun with this if you play around with the idea. If you can divide a second in half, and that 1/2 half second in half, and so on. How do we ever actually arrive at the next second in time? Same idea different application. It turns out that the smallest length of time is 3.3 x 10-44 sec (Planck's time. Your 3.3 has turned up again). We get that by measuring the amount of time it takes the fastest thing (light) to travel the smallest distance. (a Planck length) Beyond this time does not have any value.

Infinity is actually only useful in mathmatical calculations such as this. Beyond that your just blowing your mind. But it is fun to do. If people ever quit asking "what if" then we would never learn anthing new.

Infinity exists only within this Dimension, or this area of our known galaxy. So in a way it does have an end in that, it has to cross over to the next
Dimension/Universe/Galaxy or whatever you want to call it. In other words our laws of Infinity are different to those outside our 'space' leading to the conclusion that Infinity has an end.