There are different orders of infinity, but in your example both are of the same order.

Different orders of infinity are e.g. the Natural Numbers and the Rational Numbers. There is an infinite amount of Rational Numbers "between" each sequence of Natural Numbers.

I don't know much math, but I believe the relevant terms are "Cantor's Set Theorem"

First you have to understand the notion of the size of a set. The 'number of elements' definition makes less sense when considering infinite sets. Since, you know, every infinite set has an infinity of elements.

The definition of set size for finite sets is that a set is of cardinality (size) X when there exists a bijection (one to one and onto function) between it and the set of natural numbers {0,1,2,3,4,5,...,X-1}.

More generally, two sets are the same size when there is a bijection between them. One set A is larger than another B if there is a surjection from A to B but no surjection from B to A. One set A is smaller than another set B if there is an injection from A to B but no injection from B to A. One set is the same size as another if there is a bijection between them.

Under these definitions, a set is infinite just when there is a bijection from that set to a proper subset of it. So one way of showing the integers {1,2,3,4...} are infinite is to pair them up with the even numbers {2,4,6,8...}, there's a bijection between the two sets so they're the same size, but the evens are a subset of the integers, so the integers are an infinite set. So your example is wrong, the two sets you consider have the same size.

Nevertheless. The power set of a set is always of higher cardinality than the original set. That is, it is strictly larger.

If you have one infinite set, you suddenly have infinitely many infinite sets of strictly different sizes, and ordered sizes. Just take the power set of any infinite set to obtain another infinite set which is strictly larger! Each of these sets has a cardinality associated with it. So yes. There are different sizes of infinities so long as you admit the existence of an infinite set and allow the power setting operation to work on infinite sets.

Edit: if you want to go really crazy, there are infinities so big you can't actually reach them through the power set operation. :)

Different orders of infinity are e.g. the Natural Numbers and the Rational Numbers. There is an infinite amount of Rational Numbers "between" each sequence of Natural Numbers. — Echarmion

These are actually sets of the same cardinality. There's a clever geometric insight which gives you that result. Imagine a coordinate system, 2 axes. Mark every natural number pair (x,y), making a grid. You can 'snake out' from the centre and hit every pair. But this set of pairs maps onto to the set of fractions (x/y). :)

The property you're referring to is closer to the idea of the (topological) density of one set within another, in an underlying axis of real numbers, the rationals are dense (can always find rationals of arbitrary distance apart, and between every pair of real numbers), but the naturals are not. This doesn't tell you much about the cardinality of either set.

Let's take two infinite sets of numbers:
1. ...1,2,3,4,5,6,7,8,9,10,11 .....
2. .... 2,4,6,8,10,12,14 ......
So both sets follow a very obvious pattern.
Question:
Does the first set constitute a bigger infinity than the second one, as, let's take the interval (1;4) for example, the first set includes four numbers (1,2,3,4,) of this interval whereas the second one only includes two (2,4)? — Gilbert

Fdrake already answered this, but if bijections, injections and surjections aren't familiar to you, here's a one way to look at it:

Take the 1. line:

1.) 1,2,3,4,5,...

And multiply every number in that line with the number 2:

1.) 1x2, 2x2, 3x2, 4x2, 5x2,...

And you get from number line 1. number line 2:

1.) 2, 4, 6, 8, 10,...

And instead of 2, you can multiply every number with 100, which "leaves" 99 numbers between. Yet because it's an infinite set, the size is same. The trick is that if you can put the numbers in any infinite set into an order where you definately know you aren't leaving any numbers out, then that infinite set is the same size (cardinality) of the natural numbers (N=1,2,3,4,5...) because there is this mapping to them.

What Cantor did later was that he found out that not all numbers can be put into 1-to-1 with the natural numbers with his diagonal proof.

I'm assuming that this is a trivial question, but nonetheless it just occured me and I can't think of a way to approach it. If my mathematical premises are wrong (which could very well be the case) please correct me.
Let's take two infinite sets of numbers:
1. ...1,2,3,4,5,6,7,8,9,10,11 .....
2. .... 2,4,6,8,10,12,14 ......
So both sets follow a very obvious pattern.
Question:
Does the first set constitute a bigger infinity than the second one, as, let's take the interval (1;4) for example, the first set includes four numbers (1,2,3,4,) of this interval whereas the second one only includes two (2,4)?
Furthermore, applying this reasoning to the whole sequence, can the equation: Infinity (set 1)= Infinity (set 2) x 2 be infered from that?

Hey Gilbert,

I think this question has already been answered quite well by fdrake and ssu. I thought I would offer a less technical answer (though hopefully not any less correct) that might add to what has already been discussed.

I always thought of the following situation as a helpful way to understand bijections. Imagine you are setting up a large banquet. Let's further imagine that each guest gets one fork and one napkin. If I am sure to set up every seat with a fork and a napkin (no more and no less for either), then I know that the number of forks and napkins in the banquet are the same. I know this even if I don't know exactly how many forks and napkins there are. I don't need to count them, I just need to know that they are paired.

So, to your original question, is there a way to pair the numbers in line 1 with the numbers in line 2 in a similar way, i.e., in a way that I know that they are matched up exactly without needing to count any of them and being sure I've left none of them out.

There sure is! And this was alluded to in ssu's post. Let x be any number from list (1). Pair x with 2x. In this way, each number in (1) will be paired with each number in (2) and it is easy to see that it will be paired with one and only one number in (2) and that none of the numbers from the list (2) will be left unmatched. Thus, I know that the number of numbers in (1) is the same as the number of numbers in (2), in the very same way that I know I have the same number of forks as napkins at the banquet.

Hope that added something.

:point: :strong: :cheer: