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.

[Tormod]

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

[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?

[Thelonious]

Originally Posted by Turtle

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 paint job!

[zadojla]

Originally Posted by Buffy

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.