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How many different values of infinity are there?

This is not as stupid as it sounds... It can be shown that the total number of Natural numbers is the same as the total number of Integers. This seems counter-intuitive because the Natural numbers don't include negative integers, so it seems there should be twice as many integers as natural numbers, but actually there are the same number. Another strange thing is that there are the same number of rational numbers(fractions) as integers, despite the fact there are many fractions between two consecutive integers! These two strange facts can be proved by considering a one-to-one mapping between natural numbers and integers, or natural numbers and fractions - you'll never run out of natural numbers to count them. This value of infinity is called aleph-0. However, this does not work with irrational numbers such as π and root 2. There really are more irrational numbers than any other type. This value of infinity is called aleph-1 My question: are there any more values of infinity? Bunny - if you aren't interested in maths, why are you here? Becky M - I don't think infinity has a sign. Sometimes functions will zoom off towards positive infinity, and then come back from negative infinity, so maybe it wraps around? Bunny - abuse reported

Public Comments

1. none there are only different ways of expressing it.
2. Infinity!
3. No. Infinity is not a number so much as it is a concept. Therefore, there is no "value" for infinity. Infinity as a number does not exist! It is simply a "placeholder" for a number that can be approached but never reached.
4. Well, you got me with that one. Maths was never my strong point, although, sounds interesting and I'd be interested to see some decent answers...
5. The short answer is yes; there are infinities that are larger than aleph-1. Consider infinity raised to an infinite power, an infinite number of times.
6. There are no values, it descibes an idea not a particular number.
7. +oo and -oo ??
8. There are an infinite number of infinities. When you look at the cardinality of the irrational numbers, you are really looking at the Power Set of the rational numbers. Every time you look at the Power Set (i.e. the number of subsets contained in a set), the cardinality increases an order of magnitude. Thus, the Power set of a set with cardinality aleph null is aleph 1. The Power set of a set with cardinality aleph 1 is aleph 2. The Power set of a set with cardinality aleph 2 is aleph 3. Since this approach can be done as many times as there are integers, there are at least aleph null infinities. Now you need to worry about the continuum hypothesis. This talks about the existence of an infinity BETWEEN aleph null and aleph 1. As it turns out, the existence (or nonexistence) of such an infinity is independent of the axioms of set theory. Thus it may or may not exist! I hope your patience in reading this is infinite :)
9. NO
10. It's infinitesimal
11. 1. It is just a concept, not an actual number or value. If it was a value, then you could add 1 to it and make it bigger!!
12. I believe that is has been proven that the number of ways of arranging the elements in a set is larger than the set N - thus if N is an infinite set e.g. the set of natural numbers then we can repeatedly generate larger infinite sets. For good examples of infinity you could look at Hillbert's hotel and also the work of Cantor. Here seems to be an inteersting page http://www.daviddarling.info/encyclopedia/I/infinity.html
13. Infinity doesn't have a value. There are orders of infinity, however. As you noted the real numbers are a higher order of infinity than the rational numbers. If you have an infinte set and take its power set i.e. the set of all its subset, you get a set with a higher order of infinity. And you can do this, well, an infinite number of times. There is no highest order of infinity.
14. Good question. Upon real observation there are infinite possibilities of the value of infinity. In reality infinity is a concept rather than a value that we use to describe numerical or spatial concepts that are (seemingly) without end. In the most common version we think of counting numbers or the distance to the end of the universe or time, but when you include the fractional and decimal distances between whole numbers there is really no end to where it stops. things could always be measured on an infinitely larger or smaller scale.
15. Tom has it right, there is no other possible answer. Infinity is just that.