Jonsey3333

Another post inspired this question of mine (guess which post!). Is the universe infinite? What does this really mean? Does infinite really exist? If so, why are some things infinite, and some not?

-Mike

Francois Tremblay

Mathematical infinity cannot be an actuality in any form, because by definition it transcends actuality (such as no number can be higher...). It also usually entails "infinite regress" (see Zeno's Paradox for an example), which in all cases is demonstrably absurd.

Present cosmological theories estimate that the universe has existed for about 12 billion years old. Since it has been "expanding" at a finite speed also, you can see that it can't be mathematically infinite. It is, however, big enough for most purposes, including Spring Cleaning.

Using the standard definition of infinity (that is, all there can be on a given gradient or context) instead of the mathematical one, the universe is by definition infinite, although that is obviously not very informative.

[ January 29, 2002: Message edited by: Franc28 ]</p>

Clutch

Good question. Because some things are finite, and some not?

Only partly a joke. The classical definition of infinity is just: not-finite. A slightly more revealing definition is this: some collections of objects can be put into one-one correspondence with ("are the same size as") proper subsets of themselves. Those are infinite ones.

I'm not trying to baffle you with bullshit; certainly other people here know more math and can explain this clearer than me. So I won't really try, for fear of making a hash of things. But here's what I can tell you. It's worth preparing yourself from the outset for a substantial refinement of your questions, because there's a variety of well-defined conceptions of infinity. For instance:

It can have a lower bound and still be infinite.

It can have an upper bound and still be infinite.

It can have an upper and lower bound and still be infinite.

It can be infinite, and yet be smaller than many (indeed, infinitely many) other things.

These are a few truths that sit poorly with our typical intuitions about what "infinite" means. So, speaking from my own experience of learning a bit about this, be ready to have your intuitions pushed around a bit.

Theophage

An excellent post, Clutch; you are indeed a worthy poster to these boards. The definition you gave of infinity is quite correct, and your grasp of it seems spot-on.

As for the original questions by Jonsey3333:

Is the universe infinite?

I don't think so, but I don't find it logically impossible for it to be so. But this answer would make more sense in light of the answer to the next question:

What does this really mean?

This is a much better question, and one that too many people never seem to ask. People have a tendancy to blithely state that "God is infinite" or "The universe is infinite" without ever really specifying what they mean. They also tend to regard infinite things as inherently mysterious and incomprehensible, and say really stupid things like: "How can in finite mind understand the infinite?"

Don't listen to them, Jonsey, the property of infinity is understood pretty well by those who bother to learn something about it before blathering about its incomprehensibility.

A good start would be to fire up your favorite search engine (like <a href="http://www.google.com" target="_blank">Google</a>, the greatest search engine in the universe, for example) and enter the words "infinite" and "Cantor". Georg Cantor was a 17th century mathematician who is really the father of the modern understanding of all things infinite. For a limited but decent overview of his ideas, see Clutch's post.

Getting back to your question, saying that God or the Universe is "infinite" is pretty much the same as saying that God or the Universe is very. "Very what?" you might ask. And that is exactly the point.

Something isn't simply "infinite", it is infinite in a particular way. For example, you could say that the universe is infinite in spatial distance, infinite in amount of energy, or simply infinite in terms of physcially discernable states or points of reference.

Generally, when people say that the universe is infinite, I assume they are talking about spatial distance, but who knows? I personally don't think that the universe is infinite in any of the above described ways, not even an infinite number of possible spatial points between any two given points (this condition is called "discrete" or "non-continuous" spacetime)

Does infinite really exist?

You mean is there any existant thing which is infinite in some way? I peronally don't think so, but modern physics does treat some quantities as either infinitely large or infinitely small. Many sets of numbers are indeed infinite, but they aren't "real" existant things.

If so, why are some things infinite, and some not?

I've already answered no, but I thought I'd address this anyway. Why do somethings come in groups of three and some don't? I suppose the answers are similar...

Daniel "Theophage" Clark

[ January 29, 2002: Message edited by: Theophage ]</p>

Francois Tremblay

That's a novel definition, if anything, but it is probably functionally equivalent to mathematical infinity. So I'm not sure what this is supposed to bring us (and how anything can fit within this definition while not fitting in the first).

Theophage

Actually, the defintion "any set which can be put in a one-to-one correspondance with a proper subset of itself" is a definition I have seen quite a bit when talking about mathematical infinity.

Clutch

Franc, it's the standard definition of Dedekind infinity. It is informationally richer than the first one, because it specifies what criterion we're using to distinguish finite from infinite collections. Anything satisfying it will indeed "fit in" the first definition too -- but not conversely. As I understand it, you need Dedekind infinity plus the Axiom of Choice to get the classical definition.

Draygomb

I see infinity as an ever increasing number.
Infinity = (x=x+1)

Theophage

Cantor's defintion and subsequent work in the area of infinite sets is much more useful. For example, did you know that there are different sizes of infinity?

Its interesting stuff...

CardinalMan

I don't understand what you are trying to say here. How does infinity, by definition, "transcend actuality". As Theophage stated, in mathematics, there are different sizes of infinity, so some infinite numbers are larger than other infinite numbers. The whole theory of ordinal and cardinal numbers, along with transfinite arithmetic, is based on this fact. Also, very little talk of infinity in mathematics has to do with "infinite regress", and Zeno's Paradox does not lead to something demonstrably absurd, but opens up the theory of infinite series, an extremely important branch of mathematics with many concrete applications.

Keep in mind that when most people (including most philosophers) use the word infinite, they mean something much more nebulous and ill-defined than when a mathematician uses the word.

Not only are there different sizes of infinity, but there are more sizes of infinity than there are elements in any given infinite set. Just how ridiculously huge infinite sets can be, and the implications about their existence, is precisely what many set theorists study.

Actually, if a set is Dedekind infinite, then you don't need the Axiom of Choice to conclude that it is classically infinite (i.e. not finite). The converse, however, may not follow from the normal axioms of set theory without the Axiom of Choice.

CardinalMan