Jarlaxle

Thank you all for the replies. My orignal question was asking for a "real" infinty, but I am also interested in pure mathematics too. More replies / debate on the issue is greatly appreciated.

Blake

Thanks for the correction & information, CardinalMan, as well as for your appropriate reply to owleye.

Blake

ohwilleke

Do we know if there is a finite or infinite amount of matter in the Universe? Not for sure, although the current prevailing theory is that even the Universe has a finite amount of matter and energy.

Certainly, one of the most common types of infinities in daily life is the infinite number of possibilities for things. There are an infinite number of ways I can get from here to there. There are an infinite number of possible weather states tommorrow. There are an infinite number of ways to arrange paper in your office.

A good example of an infinity we use in daily life is radio. Radio waves useful because they go infinitely great distances in a very predictable way that is infinitely smooth. (Quantum mechanics muddies this, but only by positing particles that can be infinitely many places in each succeeding moment).

Infinities don't have to be "big" in a physical sense. A mile of shoreline can have infinite, or near infinite detail.

Another funny thing about infinities is to recognize that infinites can be bounded, and indeed generally are. "Infinite" and "all encompassing" do not mean the same thing. For instance, the infinite number of ways to arrange paper in your office may have no impact on what goes on in the next door over. Intergers are just a small subset of all numbers, even though it is an infinite set of numbers. The infinite number of points between your fingers are never the less only an inch long taken together.

Calculus, and its cousins, real and complex analysis, are fundamentally studies about how to summarize infinitely detailed things in finite ways through cleverness. For example, one of the classic problems of complex analysis (the calculus of imaginary numbers) is essentially equivalent to the problem of measuring the finite weight of an infinitely dense black hole (using a mathematical method called a "path integral").

One of the other funny things you discover in math is that infinities (i.e. continuous functions) are often much easier to deal with than finite quanities (i.e. discrete functions) . . . because any useful infinite quanity can be summed up rather neatly, while a whole host of finite particularities, must generally be described on a more extensive individualized basis to be fully accurate. Smoothness, regularity and infinity are intimately related notions, while jaggedness, irregularity and finiteness are their oppposites.

Jarlaxle

What does "near infinite" mean?

From what I have seen so far, there are different "levels" of infinity. But how much space is there between one infinity and the next larger infinity? How much space is there between the first level of infinity and "near infinity"?

My brain is exploding

ohwilleke

[QUOTE]Originally posted by Jarlaxle:
<strong> What does "near infinite" mean? </strong>

Near infinite means incapable of being counted, but may or may not be finite.

For example, the number of digits in "pi" in a base 10 number system may or may not be finite. We've taken it out to billions of digits without getting an exact decimal representation yet, but we have no theoretical way of proving that pi doesn't have exactly 1 trillion and 42 digits in base ten. The number of digits of pi in a base ten number system is "nearly infinite"

Similiarly, it is possible that at some point the trillions and trillions of twists and turns in a shoreline resolve into a finite number of straight lines, perhaps at a subatomic scale. But, it is also possible that the shoreline gets more and more detailed and never resolves with what we thought was a subatomic scale breaking down into a string scale breaking down into even smaller and smaller subcomponents of an infinitely complex design.

<strong> But how much space is there between one infinity and the next larger infinity? How much space is there between the first level of infinity and "near infinity"? </strong>

Infinity is not a place. Infinity is basically an "un-number", in a bit the way that black and white are "un-colors". All infinite numbers of things share the basic feature that no matter how hard you try, it is impossible to count them all sequentially.

"Levels" of infinity is a way of saying that some infinite numbers of things that have qualitative similarities. I think it is more useful to think of kinds of "infinities" than levels of infinities when you are first trying to understand the concept. You make rules that separate different infinite groups of things into categories based on if they fit certain rules. For example, if you know two members of a group, are there an infinite or finite number of members between them. Or, do the members of the group have any sequential order or not?

For example, your basic "whole numbers" infinite set is what you might call a "one ended" infinity, in that you can list every number in the group from point A to point B. It is comprised of enumerable finite ranges of numbers.

The infinite set of "real numbers", in contrast, can't be listed in order. No matter what two numbers you list, there is always a number between them.

The infinite set of complex numbers doesn't even have a clear "order", it is a two dimensional infinite set, rather than a one dimensional infinite set like a number line.

Once you break the different kinds of infinite groups of things into several categories, you can, if you choose define a method for thinking of one as larger and one is smaller using a rule like: "If you can match up infinite groups on a one to one basis, then they are equal in size; if you can match up members of a first group to members of a second group but not visa versa, then the second one is bigger."

These measures of size make theoretical sense, but not intuitive sense, because no infinite group is "bigger" than the other in the narrow arithematic measure of bigger meaning having an absolute number of members bigger than the other group. For example: The number of real numbers between 1 and 2, is considered "bigger" in a math sense, than the number of whole numbers, even though the latter group "takes more space" the way we usually think about it.

[ April 23, 2002: Message edited by: ohwilleke ]</p>

CardinalMan

Actually, we can prove that pi is an irrational number (can not be expressed as a fraction a/b), and hence the decimal expansion of pi will never repeat. In other words, we will never come to point where we see 427427427... repeating forever.

When listing a set, you do not have use the standard ordering. Between any two rational (fractional) numbers lies another, yet you can list them nicely one after another so that you reach each one at some finite time. Of course, this ordering differs from the usual ordering of the rational numbers.

In fact, if you accept the Axiom of Choice, then one can order any set, including the set of real numbers, in a sequentially ordered list. The only catch is that you may have to have "limit" stages. In order words, we can start listing real numbers, say 1,1.3,pi,2.89,e,... and after we list countably many, we haven't exhausted all real numbers, so we start again. For example, another way to "list" the integers is 0,1,2,3,... -1,-2,-3,.... In this more general type of list, the number -1 is above infinitely many numbers.

The same argument applies to the complex numbers.

I'm still confused by your idea of "near infinite", at least within mathematics. Certainly, in the standard viewpoint of mathematics, every set is either infinite or finite, regardless of whether we know the answer. There are some philosophies of mathematics that deny this fact, but I'm not sure if this is what you are arguing.

CardinalMan

owleye

CardinalMan...

I appreciate your reply, of course, if the topic was mathematical. Apparently Jarlaxle had only a passing interest in "real infinities" and seemed to find your post more informative than mine.

In any case, on the quesiton of whether it is important to get at the concept of infinity itself before engaging in the question of this topic, I have my doubts, since the common notion, meaning: not bounded by any finite number (i.e., not finite), is understandable for the purpose of pursuing the question of whether or not there are an infinite number of things of any kind at all in the world quite apart from the further distinctions made by modern mathematics.

Notwithstanding, I would agree that physicists have benefited from the prior development of a variety of mathematical models of infinity (though their use is often negative -- i.e., they often run from it, for example, through such techniques as renormalization), and, quite a part from their immediate application, infinities are worth pursuing. Moreover, since much of mathematics has been formalized, there is some philosophical benefit to analyzing quantificational concepts, such as infinity, so that they might serve some epistemological or metaphysical purpose. If this is the topic, I'd be only too interested. Finally, as evidence that the issue identified by the topic is not without some philosophical interest in itself, I give you the Intuitionists (who deny completed infinities). They have made significant inroads in identifying the lines of demarcation that distinguish themselves from Platonists (such as Cantor and Godel).

owleye