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# How big an infinity?

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It's funny to think that one infinity can be bigger than another infinity, but it's true.

The number of possible numbers is obviously infinite. The number of prime numbers (2, 3, 5, 7, 11, 13, 17...) is also infinite. But how much bigger an infinity is the number of total numbers compared to the number of prime numbers?

I have no idea..?

thats a question best asked to a pot head when he just got high.

Fredrik said:

Infinity is not a number, so one would think the whole concept of bigger/smaller is lost on it.

heh, infinity is just assigned to something that's impossible to count because there's no limit to it. there's no real way to compare different degrees of infinity because they're all infinite by definition. Even every possible decimal between 0 and 1 is infinite, so every single number in there is < 1 but it's still infinite

what's really freaky is when you get stuff like (1/infinity) = 0 or how with logarithmic curves you can have an infinite amount of numbers but they add up to a finite amount. For instance 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... = 1 even though the series goes on forever it's still equal to a finite number. FREAKY MAN.

There's no way to define the size of an infinity. The closest you can get is cardinality, which is a property of sets.

The cardinality is the same for two sets if you can pair them bijectively. For example, [1,2,3,4] and [2,4,6,8] have the same cardinality (4), because you can construct a function that generates set A from B and vice versa.

As you can pair each integer with one prime number this way, and the sets' cardinalities therefore are the same, the infinitenesses of all prime numbers and all integers are the same.

However, the set of all real numbers cannot be mapped to bijectively to the set of all integers. The set of all real numbers has a higher cardinality - its infiniteness is bigger.

I am scared. I actually understood what Fredrik said.

There are such things as classes of infinity, but I found it so boring and it's so many years ago that I've forgotten pretty much all the details. :)

I think "aleph zero" was the term for a countably infinite set (e.g. all integers, all prime numbers, etc.)

There are also uncountable infinities and stuff.

Actually (now that I've clicked on it), Fredrik's link seems a better source than my crappy memory.

Regarding how many prime numbers there are in any given range of positive integers, there are some formulae/limits/approximations for this, but they have nothing to do with infinity. Prime Number Theorem

BTW, the Continuum Hypothesis is relevant to this.

Alot of people mistake the difference between eternity and infinity.

I was always told that infinity was as high as you yourself could count. Don't know though.

Melfice said:

Alot of people mistake the difference between eternity and infinity.

Someone is bound to come along and mention serenity, so I'll get in first.

I've always heard that infinity goes on forever, but what I do know.

Melfice said:

Alot of people mistake the difference between eternity and infinity.

I am happy to report that I have never mistaken the difference between Quasar's source port and a mathematical concept.

Here's the way I see it: Yes, but on an infinite scale.

Consider the speed at which each set of infinite numbers is growing. That speed itself is infinite. And of course, that infinite speed is growing at it's own infinite speed.

So, we have an infinite loop (or tree) of speeds controlling the speed at which an infinite set grows. An infinite loop of infinity controlling infinity's speed, essentially.

Or, look at it like this:

Infinity1 grows at the speed of Infinity2
Infinity2 grows at the speed of Infinity3
Each infinity grows at the speed of another infinity, so you enter a logic loop of thought about this that never ends, thus creating another overlaying infinity over all that, which is an endless number of loops controlling an endless number of loops.

Go watch "Escape From The Planet Of The Apes" for more info and a comparison using a painter painting a picture of the painter painting the picture of the painter painting the landscape, so on and so forth. That loops controlled by another infinity.

Then again, I've never got past Algebra 2.

{edit)
Actually...I guess you can never make a true statement of infinity. Saying that infinity is infinitly big and has an infinite number of layers controlling it is leaving off an infinite number of layers of description. You aren't describing some of those "infinities". We can't talk correctly of infinity, except by maybe saying "42", heh

TeamKill said:

Go watch "Escape From The Planet Of The Apes" for more info and a comparison using a painter painting a picture of the painter painting the picture of the painter painting the landscape, so on and so forth. That loops controlled by another infinity.

Or you could just watch Spaceballs. :P

I love the paradox that there are exactly as many even numbers (2, 4, 6, etc) as there are whole numbers (1, 2, 3, etc). Thinking about it you would think "uhh wouldnt there be twice as many whole numbers" but if you take that infinite list of whole numbers and multiply each number by 2, you have the list of even numbers. CRAZY!!!

uh, wtf are you talking about. infinity is infinity. both prime and natural numbers go on forever, and although the list of natural numbers would have some in its list you wouldn't find in a list of prime numbers, they both go on forever and thus are equally long. wtf were you smoking to think this idea made any sense?

The reals and the naturals both "go on forever", but they are not the same order of infinity.

AndrewB referred to "prime numbers" and "all possible numbers", remember.

it doesn't matter though. as both sets never end, they are both infinitely long; just because you skip more numbers in a set of prime numbers than in a set of all numbers doesn't make one infinity larger than the other, it just means one skips around more. if the sets were to a million, yes, the set of all numbers would have more entries.

but infinity is not a variable. it means the sets keep on going and going, as they can. infinity is not 'the last number', there is no 'last' number. it merely represents the principle that numbers never end. while the nth variable of a set of prime numbers will obviously be higher than a the nth variable of a set of all numbers, both sets are still infinitely large. c'mon guys, this is common sense =P

Barbie said:

math is hard

I accept the conclusion that all infinites are equal in their infiniteness.

But the question is, as the list of prime numbers and whole numbers approaches infinity, what does the ratio of whole:prime approach? Pi? Infinity? 2:1?

AndrewB: see the link I gave above for Prime Number Theorem. Very brief summary: the logarithmic integral function provides a good approximation to the number of prime numbers below any particular number. A cruder approximation (but one that is easier for a pocket calculator) is n/(ln n). Oh, and with your conclusion "all infinites are equal in their infiniteness" you are much further from the truth than you were in your original statement at the start of this thread.

Baldy/Grogglo: see the link Fredrik gave, and the one I gave for Continuum Hypothesis. Not all infinities are the same. Just for clarity: the set of prime numbers and the set of natural numbers have the same cardinality. The set of real numbers has a different cardinality.

Fun with math!

The largest known prime is 2^13466917-1 which is 4,053,946 digits long.
The second largest is 2^6972593-1 and is 2,098,960 digits.

Clicky.

Wtf? Obviously they are both equally infinite. One is simply more occurant.

Note that AndrewB's original post referred to "possible numbers". That's a bit imprecise. He may have had natural numbers in mind, but "all possible numbers" includes all real numbers. And they're not countable.

This thread is infinitely stupid :P

And yeah, as people have said, infinity has no size, so comparing it to another can't happen.

mmnpsrsoskl said:

And yeah, as people have said, infinity has no size, so comparing it to another can't happen.

I imagine that's a matter of opinion. I've seen the "one infinity larger than another" conversation on a few math sites.

Grazza said:

Not all infinities are the same. Just for clarity: the set of prime numbers and the set of natural numbers have the same cardinality. The set of real numbers has a different cardinality.

What he said.

huh...scratches my head wtf is goin on??