What is ∞ + ∞ = ????

[polymath257]

If you add ∞ to ∞, then what do you get?...

The quick answer is that it depends.



There are several different types of infinity in mathematics (and I won't even go into philosophy).

1. Limits

Limits are ways of describing what happens to a variable when another variable gets closer and closer to a value without getting there, or when the second variable gets larger and larger.

So, for example, if you let y=(x+1)/x, as x gets larger and larger, y gets closer and closer to 1. We say that the limit of y as x approaches infinity is 1.

Now, sometimes the second variable gets larger and larger. For example, as x gets close to 0, y= 1/x^2 gets larger and larger. In this case, we say that the limit
of y as x approaches 0 is infinite.

Now, suppose that the limit of y as x approaches a number is infinite and, similarly, the limit of z as x approaches that value is also infinite. Then the limit of ywill also be infinite.

So, in this case, ∞ + ∞=∞.

But, it is crucial to remember that ∞ is not a number in this. It is simply a short-hand for saying that something gets larger and larger without bound.

2. Cardinality (sizes of sets)

A set is a collection of mathematical objects. Anything in the collection is called a member of that set. We describe sets by listing the members, separated by commas, between braces. So, A={1,2,3,4} is a set with members 1 and 2 and 3 and 4.

We say two sets have the same cardinality if they can be paired off in some way, with each member of one being paired with exactly one member of the other.
For example, the sets A={1,2,3,4} and B={5,7,9,11} have the same cardinality because we can pair 1<->5, 2<->7, 3<->9, 4<->11. This is not the only way to pair.
But we only need one way to say say the sets are the same cardinality.

Intuitively, cardinality corresponds to the size of the set, so sets with the same cardinality are of the same size.

Now, a natural number is an ordinary counting number, like 1,2,3,4,5,6,... We can look at the set of ALL natural numbers,
N={1,2,3,4,5,6.....}.

But this set has a strange property: we can pair it off with the (smaller?) subset of even natural numbers E={2,4,6,8,10,12,...} as follows:
1<->2
2<->4
3<->6
4<->8
etc.

In this, every natural number on the left is paired with exactly one even number on the right. So, in this formulation the set of natural numbers has the 'same cardinality'
as the set of even numbers, even though the set of even numbers is 'smaller' in a very real sense.

This property of having the same cardinality as a proper subset is characteristic of infinite sets. And the set of natural numbers is an infinite set.

There are many other sets that have the same cardinality as the set of natural numbers: the set of integers (both positive and negative counting numbers), the set of all fractions, and many others.

But, it turns out that if you look at the set of *all* decimal numbers, that is a larger cardinality! if you allow for infinite, non-repeating decimal numbers (like pi), there is no way to make a pairing to the natural numbers!

There is more than one 'size' of infinity!

None the less, if you take two infinite sets and 'add them' together, the result is another infinite set (of the larger cardinality), so in this sense ∞ + ∞=∞, although
we can do a more refined formula detailing the cardinalities.

3. I'll save ordinals for another post.