Thinking about infinity

[Alex K]

There is bigger numbers than infinity

"infinity" is not really a technical term for any particular (cardinal) number. If we are talking sizes of sets, both the natural and the real numbers have "infinite" size, but they are different infinities. The latter is larger than the former.

[robvalue]

Just as an aside:

When we use the concept of infinity in mathematics, it is usually employed as a limit. We allow a number to tend towards infinity, and we see what effect that has.

For example, 1/n tends towards zero as n tends towards infinity.

The technical way of writing this is:

For any positive number k, we can find a number m so that 1/n < k for all n >= m (greater than or equal to)

[In general we would use |1/n| which means to ignore any negative values, but we know in this case, dealing with positive values of n, 1/n is always positive anyway.]

In other words, however small we want the value of 1/n to be, we can find a point so that the values of 1/n that follow are all below that number. It gets arbitrarily close to zero, and so we say that it becomes zero as the limit is taken of n=> infinity.

It's important that all values of 1/n are below this value of k after this point we find, or else we might be dealing with an oscillating function that jumps in and out of the required zone.

This value of m can be easily found:

We want 1/n < k

Multiply by n to give 1 < nk

Divide by k to give 1/k < n

Turning that around gives n > 1/k

So the value of m will be the first whole number greater than 1/k. Let's say k = 0.000001 (a millionth). Then we need

n > 1/0.000001

n > 1,000,000

So 1/n < 0.000001 for all n >= 1,000,001

Did that make any sense to those not familiar with the subject?

[Wryetui]

I have a question that maybe is not related directly with the OP. Can the concept of infinity exist in nature, that is, can we observe infinity in nature? I am asking this because I have witnessed atheists talk about the infinite and I do not believe naturalism and infinity are things that can co along.

[bennyboy]

I have a question that maybe is not related directly with the OP. Can the concept of infinity exist in nature, that is, can we observe infinity in nature? I am asking this because I have witnessed atheists talk about the infinite and I do not believe naturalism and infinity are things that can co along.

Why not?

Why do you think that the God idea solves any philosophical issue?  What about God makes infinity seem more sensible except a question-begging definition: "God is He who exists infinitely while taking an interest in finite beings" or whatever?

Here's the typical Christian argument:
C: Infinity is impossible.  We need something outside of reality who is infinite, and only with Him is true infinity possible.
A: That's special pleading.  If you want to say the impossible is possible, why not just attribute that "impossible" property to what we know exists?
C: Cuz. . . it's impossible.

[robvalue]

Maybe our reality is infinite, in one respect or another. It's logically consistent. Infinite time, infinite length... whatever. Or infinite in some other way we haven't even comprehended yet.

Maybe things can be theoretically split into infinitely many pieces.

Excluding extremes like this is not generally what science is about. All we can do is observe, test and model. We can't make sweeping statements that apply to all of reality and beyond, just because we say so, based on our current best models of the areas available for testing.

If someone claims infinities are impossible, the burden of proof is on them to back this up. And I'd say that is an impossible task, at least currently. "Don't make no sense" isn't adequate.

[Alex K]

I have a question that maybe is not related directly with the OP. Can the concept of infinity exist in nature, that is, can we observe infinity in nature? I am asking this because I have witnessed atheists talk about the infinite and I do not believe naturalism and infinity are things that can co along.

I can't think of anything that would allow us to observe some form of infinity.

[robvalue]

Good point.

But whether we can observe it, and whether it exists, are two different questions.

[ErGingerbreadMandude]

There is bigger numbers than infinity

"infinity" is not really a technical term for any particular (cardinal) number. If we are talking sizes of sets, both the natural and the real numbers have "infinite" size, but they are different infinities. The latter is larger than the former.

I wouldn't say larger, more like denser.
Like when I think about comparing infinites, I cut up both the infinites equally. Like,set of natural numbers cut up at 5. And the set of real numbers cut up at 5. So when we look at it the cut up version of the real numbers will have more numbers than the cut up version of natural numbers.
Like the cut up version of natural numbers will only have 1,2,3,4,5 whereas the cut up version of real numbers will have 1,2,3,4,5 and ever other number between them. So I think all infinities are equal in size but different in density. Like according to my example set of real numbers and natural numbers would be equal in size but different in density, like the set of real numbers would be infinitely denser than set of natural numbers.
Does that make any sense? Lol

[ErGingerbreadMandude]

Good point.

But whether we can observe it, and whether it exists, are two different questions.

What about a circle?

[robvalue]

As in, infinitely many "sides"?

Abstractly, it can be viewed that way. Whether such a thing can exist in reality is another question. Maybe, maybe not. If you zoom in far enough, will it ever stop being a continuous, perfectly smooth circle? My instinct says that it probably will stop, at some point. But I can't know for sure.