Infinity

[Tormod]

Okay, here is a topic I've been thinking about for a long time. It also happens that I am about to start reading John Barrow's latest book about infinities.

 

There was a recent post about what you get when you divide infinity with itself. My response to that was that there are many kinds of infinities, and they have different properties.

 

For example, we assume that there are an infinite amount of positive and negative integers. Whatever number x you can think of, there is always x+1 so it will go on forever.

 

But this type of infinity is not "physical". Now, if we take a rubber band, it is very physical. It is bounded in that it has a finite surface area, yet an ant can walk on it forever without coming to an end. This is also a kind of infinity, but it is a semantic infinity because it depends on our definition of an "end".

 

So...what would REAL infinities be like?

 

Consider time. If we say that time (as we know it) started with the Big Bang, then it is not infinite in the direction of the past. But we do not know if it is infinite as we move towards the future. We can make different theories about how the universe will end, but we cannot ever test them so we might never know for sure. Can we then say that time is infinite?

 

The same reasoning goes for the universe. If it is infinite in expanse, then how can it have started with the big bang? Can an infinity be created in a finite amount of time (ie, 13,7 billion years)? Or can the Big Bang still be the start of our universe even if it is currently infinite? (This is not intended as an argument against the Big Bang theory, which I currently subscribe to).

 

Okay, no real question popping into my mind, except this: Which kidns of infinites are there, and what are their properties?

[Fishteacher73]

With the concept of multidimensional theories (ie M Sting theory, etc.) the idea of wraping multiple dimensions onto our "standard 4" has become a bit more common. We have exponential growth inward. Similar to a mathamatically infinite number of numbers just between 0 and 1. By doing this, can we therefore create and infinite space "inside" a finite one (beyond just in the mathmatical sense of the word infinite)?

 

Could it be that infinity is actually an "inward" progression as opposed to the concept of this never ending expanse spreading "outward" forever?

 

Perhaps this is just ramblings and the math or physics really do not support such a notion (Although I am not really aware of anything that contradicts these ideas, but I'm not well read on multi-dimensional physics or mathmatics).

[Buffy]

We need Turtle here probably. Its been so long so I'm rusty but the mathematical distinctions on infinities are well defined. I remember the main one being, think of the infinity represented by integers (as Tormod mentioned above), then think about real numbers: you've got an infinity of numbers *between* each integer. There are others as well. The integers/real numbers one is interesting because its got lots of real world applications, like (to again bring Turtle in here) Zeno's paradox (do "I feel Lucky" when Googling and you'll get: http://planetmath.org/encyclopedia/ZenosParadox.html).

 

I'll be back on this one. Its interesting both mathematically and philosophically.

 

Cheers,

Buffy

[Turtle]

I'm blushing, Buffy. You are too kind.

Tormod said "Okay, no real question popping into my mind, except this: Which kidns of infinites are there, and what are their properties?"

 

Answer: I think I said in the other thread something to the effect that infinity is growth without bound. Taking that view, one "measures" an infinity based on it's beginning, since you can never reach the end. As Tormod pointed out you can always add 1 more.

So, I'll take the Strange Numbers I have been discussing as an example of different "size" infinities. The set of integers is the first infinite series in the experiment & one might say they are close-packed;no gaps (no fractions). Now from that set of integers I found Strange Numbers, also an infinite set. Now the first Strange Number is 24 & the next 30, etc. & you see they start late & are not close-packed.

The upshot is the infinity of integers is greater/bigger than the infinity of Strange Numbers. In a similar fashion if you make your integer number line into a Real Number line, the infinity of Real Numbers is larger than the infinity of integers.

I think if you consider "infinity" a verb & not a noun, you have a better idea what we are trying to get out. :cup:

[Tormod]

Yeah, yeah, yeah. All good responses. But I want to see more kinds of infinities than endless rows of numbers. :cup:

[Turtle]

Ok, I'll try this. I suggested Infinity is a verb. If it is so, then it is action which implies movement which implies velocity, implies speed. So if you want to compare some infinitys, rather than say one is larger or smaller, say one is faster or slower than another? :cup:

[Thelonious]

I suggested Infinity is a verb. If it is so, then it is action which implies movement which implies velocity, implies speed. So if you want to compare some infinitys, rather than say one is larger or smaller, say one is faster or slower than another?

 

One know this to be true. For instance:

 

lim 2x/x

x->∞

 

In this case, one notes that, while both the numerator and denominator are infinite, the numerator is increasing twice as fast as the denominator. So, yes, there is a speed at which infinity does increase, and a velocity if one moves into the realm of integers.

[Turtle]

Excellent. So now that we have speed, let's figure out how to hold some kind of race! Tormod's Rubber Band Boat Race. Function against function! Hull design against hull design. Obviously it's no contest with the functions(engines?) you brought Thelonius, so we need to define some classes somehow. We'll need a course too.

I'm already distracted by thinking up a bright :cup: paint job!

[zadojla]

We need Turtle here probably. Its been so long so I'm rusty but the mathematical distinctions on infinities are well defined.

I can't check what I'm about to say because accessing those books would disturb my sleeping family, but I think Cantor came up with classes of infinity, which he signified with an aleph. Aleph null was the infinity of rational numbers, I think aleph one was irrational numbers, and aleph two was the infinity of the number of points in a line segment. Each class was infinite, but also infinitely larger than the lower classes. There was no aleph three, as I recall.

[Turtle]

That all sounds right about Cantor; I recall Cantor"dust" in that regard. I keep toying with how we'd make any kind of race fair; it doesn't look good. :cup:

[C1ay]

Now, if we take a rubber band, it is very physical. It is bounded in that it has a finite surface area, yet an ant can walk on it forever without coming to an end.

 

Perhaps he would like to take such an endless stroll on my avatar :)

 

So...what would REAL infinities be like?

 

One of my favorites is the infinite set of infinite sets that lie on the number line between 0 and 1. The first of the sets is the never ending sequence 1/2, 1/3, 1/4, 1/5.... that is the infinite set between 0 and 1/2. The second set is 2/3, 2/4, 2/5, 2/6, 2/7.... which is the infinite set between 0 and 2/3. As the numerator n is indexed for each set where the denominator d begins at n + 1 we have an infinite number of sets. For me this helps to illustrate the fact that for any 2 points you can pick on the number line there are an infinite number of infinite sets that lie between them. This is nature, infinite sets of infinite sets.

 

Consider time. If we say that time (as we know it) started with the Big Bang, then it is not infinite in the direction of the past. But we do not know if it is infinite as we move towards the future. We can make different theories about how the universe will end, but we cannot ever test them so we might never know for sure. Can we then say that time is infinite?

 

The same reasoning goes for the universe. If it is infinite in expanse, then how can it have started with the big bang? Can an infinity be created in a finite amount of time (ie, 13,7 billion years)? Or can the Big Bang still be the start of our universe even if it is currently infinite? (This is not intended as an argument against the Big Bang theory, which I currently subscribe to).

 

I don't think time started with the big bang, only our relative concept of time. I think of the universe that resulted from the big bang as being one of many in a larger infinite space. Imagine the room where you sit as representing this infinite space. Now hold in your hand a microscopic soccer ball to represent the universe as we know it, that occured from the big bang. Compared to the infinite space around it, it is infinitesimally small, yet to us it is infinitely large, a finite universe in infinite space. Imagine many big bangs in this infinite space creating many finite universes like our own. Some of these will have occured long before ours and many are yet to come. In this space the time is not relative to our own big bang, it is infinite.

 

One thing to ponder, if our universe is constantly expanding, what is it expanding into? What surrounds it? For me the answer is an infinite space which surrounds it. A space which likely contains many finite universes as we think of them. It is filled with matter from many previous big bangs. Some of this matter may have even occupied the vicinity of our own big bang and become part of our own universe. This would explain the globular clusters in our own universe that appear to be 15 - 20 billion years old even though our own universe is only 10 - 14 billion years old.

[Queso]

One thing to ponder, if our universe is constantly expanding, what is it expanding into? What surrounds it?

I love stephan hawkings theory on this. He described our universe as a bubble in a pot of boiling hot water. The collision of two bubbles would be catastrophic.

[Tormod]

Some of this matter may have even occupied the vicinity of our own big bang and become part of our own universe. This would explain the globular clusters in our own universe that appear to be 15 - 20 billion years old even though our own universe is only 10 - 14 billion years old.

 

I am not aware of any such observations that have not been corrected to match the perceived age of our universe. I will happily look at any such evidence.

 

If our universe expands into "space", then it is by definition boundless. However, the "space" it expands into must be something else than the space in our own universe, because the properties of our universe are direct results of "our" big bang. I assume that no two big bangs would produce the same outcome, since the sheer amount of events happening at extremely short times are not likely to be repeated in the exact same way twice.

 

So while this is an interesting way to portrait infinity, it uses something outside our own universe as an explanation. I am asking for examples of physical infinities within our universe! :)

[pgrmdave]

I don't know that there can be an infinite direction without some kind of curvature of the direction. For example, I can travel in any straight direction on earth(assuming it was a flat sphere) without stopping, but that is only because earth is curved. Infinity only really exists as a concept, I think.

[Fishteacher73]

Would it be possible to be infinite in one spatial axis but bound in the others(such as a vertical or horizontal line or ray), or is to imply infinity, applies it in all directions?

[Tormod]

These are very good questions. I don't have the answers. But let us for a moment assume that the universe has a finite past but is open and will never stop expanding.

 

What does that say about the direction of expansion? Is it an infinite line in the nth dimension? Will it necessarily be curved in the xth dimension? Because what is curved in 3 dimensions may be flat in 2 (like a sphere's shadow on a piece of paper).

 

And what does it say about this kind of infinity? It is a physical infinty because it is related to the cosmos itself. It also is finite in one direction, but infinite in another.

 

Do you think this is possible?

[Tormod]

Another type of infinity we often hear about is the singularity that resides inside black holes. They are said to be objects of infinitely small sizes with infinite mass.

 

The problem seems to me that there is no way to measure this as of yet. Still, we expect to find such a singularity at the beginning of the universes's timeline (except in superstring theory).

 

I wonder - can these objects really exist? How can something achieve infinite mass when (for example) it is impossible to reach infinite speeds?