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Infinity Plus Three (Maths For Normal People III)

Are there more even numbers than odd numbers? How much can we take away from infinity to keep it infinite? And how much is “ten times infinity”? Is it more than “twenty times infinity”? Do these questions have a meaning at all? Let us delve into the depths of the infinite oceans once again!

Infinite sets have remarkable properties. We shall examine some of them in this and the following article.

How Many Even Numbers Do You Know?

For instance, let me ask you how many even numbers exist? You will probably answer that one half of natural numbers are odd and one half even, so the number of even numbers is one half the number of all numbers. Sounds reasonable? It would, if we weren’t dealing with infinite sets.

I shall quote from my set cardinality article:

Two sets have the same cardinality if, and only if, all their members can be put into a one-to-one correspondence (mathematicians say that there exists a bijection between the sets). This means that to every member from the first set we can assign one member from the second set.

Using this definition, we can prove that there as many even numbers as there are all natural numbers! In other words, to every natural number like 1, 2, 3, 4, 5,… we can assign an even number like 2, 4, 6, 8, 10,… How do we do that? Very simply:

To every natural number we assign its double

To every natural number we assign its double, which is always even. The correct mathematical notation is surprisingly not a picture, but “<$ n~\mapsto~2n $>”, meaning “to every natural number <$n$> assign its double, <$2n$>.” With specific numbers it looks like this:

<$$ \displaylines{ 1 \mapsto 2, \\ 2 \mapsto 4, \\ 3 \mapsto 6, \\ \vdots } $$>

In this way, every natural number has its counterpart in the set of even numbers. Conversely, every even number has its counterpart in the set of all natural numbers, since we could just as well define the one-to-one relation for even numbers as <$ n~\mapsto~\frac{n}{2} $>, meaning “to every even number <$n$> assign its half, <$\frac{n}{2}$>,” or

<$$ \displaylines{ 2 \mapsto 1, \\ 4 \mapsto 2, \\ 6 \mapsto 3, \\ \vdots } $$>

We have found a one-to-one relation between <$\{1,2,3,\ldots\}$> and <$\{2,4,6,\ldots\}$> which exhausts all members of the sets (remember that this relation is called a bijection). Therefore, the set of even natural numbers has the same size as the set of all natural numbers. Precisely stated, <$|\mathbb{N}| = |\{2,4,6,8,\ldots\}| = \aleph_0$>.

Does it look like I am cheating? How could this ever work? Surely we must run out of numbers somewhere? Well, I challenge you to prove me wrong :-). Try to give me a natural number so big that I can’t find its double! Or try to give me an even number such that I can’t halve it!

Infinity, Give Or Take

Let’s now take a look at more generic modifications of an infinite set. What if we added some elements to <$\mathbb{N}$>? For instance, consider the set <$A = \{a,b,c,1,2,3,4,5,\ldots\}$>. How big is it?

The answer is, perhaps not so surprisingly, <$\aleph_0$> again. We assign 1, 2, and 3 to the three letters and then just put the numbers from 4 on into correspondence with the numbers from 1 on.

Bijection between A and natural numbers

We could write this bijection “mathematically” as

<$$ \displaylines{ 1 \mapsto a, \\ 2 \mapsto b, \\ 3 \mapsto c, \\ 4 \mapsto 1, \\ 5 \mapsto 2, \\ 6 \mapsto 3, \\ \vdots } $$>

The general rule is that adding a finite number of items to an infinite set doesn’t change its size. In terms of set cardinality,

<$$ \aleph_0 + n = \aleph_0\; \text{for every natural number } n. $$>

Removing a few items from an infinite set doesn’t change its size either. For instance, let <$B = \{100,101,102,\ldots\}$>. We can again find a bijection between <$B$> and <$\mathbb{N}$>:

Bijection between B and natural numbers

In terms of set cardinality,

<$$ \aleph_0 - n = \aleph_0\; \text{for every natural number } n. $$>

But wait! We have been examining the set of all even natural numbers and concluded that its cardinality is also <$\aleph_0$>. How do you construct this set? You take the set of all natural numbers and remove the odd numbers from it. But there are infinitely many odd numbers! We have removed infinitely many members from a set and it still retained its infinite size!

Since odd numbers constitute one half of <$\mathbb{N}$>, we can infer that halving an infinite set doesn’t change its size, or <$\frac{\aleph_0}{2} = \aleph_0$>. In fact, we could be more daring and remove even more numbers. If we keep every fifth number and throw away all other, we create a set <$\{5,10,15,20,\ldots\}$> that still has cardinality <$\aleph_0$>! (The bijection here is <$n \mapsto 5n$>.) If we kept only every billionth number, the cardinality still wouldn’t change. In general, we find that

<$$ \frac{\aleph_0}{n} = \aleph_0\; \text{for every natural number } n. $$>

Note that this little experiment also shows that <$\aleph_0 \cdot n = \aleph_0$>. If we take our set <$\{5,10,15,20,\ldots\}$> and put all the “missing” numbers back in, we increase its size tenfold… well, not exactly increase, since <$|\{5,10,15,20,\ldots\}| = \aleph_0$> and <$|\{1,2,3,4,{\bf 5},6,7,8,9,{\bf 10},11,\ldots\}| = 10\cdot\aleph_0 = \aleph_0$>.

Infinities Don’t Like To Be Subtracted

We have to be careful with all this removing of numbers. What happens when we remove all numbers starting from 100 from <$\mathbb{N}$>? We get this pathetic set <$\{1,2,3,\ldots,99\}$> which couldn’t be infinite even in your wildest dreams.

Since we removed the set <$\{100,101,102,\ldots\}$> with cardinality <$\aleph_0$> from the set <$\mathbb{N}$> also having cardinality <$\aleph_0$> and got the set <$\{1,2,3,\ldots,99\}$> with cardinality 99, we derive that

<$$ \aleph_0 - \aleph_0 = 99. $$>

Right? But hey, what if we removed all numbers from 1,000 onwards? We would be left with the set <$\{1,2,3,\ldots,999\}$> and, by the same argument,

<$$ \aleph_0 - \aleph_0 = 999. $$>

We could even remove all numbers from <$\mathbb{N}$> and obtain

<$$ \aleph_0 - \aleph_0 = 0. $$>

This doesn’t seem quite right, does it? Because of fishy results like these, we forbid subtracting infinities in mathematics. It is actually quite simple – we just say that the operation “<$\aleph_0 - \aleph_0$>” is not defined, and woe befall anyone who tries to do that!

This decision is well grounded, not arbitrary. The problem here is that “<$\aleph_0 - \aleph_0$>” doesn’t have a clear result. Depending on the specific sets concerned, the result of the subtraction could be any finite number, or even <$\aleph_0$> again (as we have seen in the previous sections).

Note well that this doesn’t mean we can’t remove infinite parts of infinite sets, only that we have to be especially careful when handling set cardinalities, which is what <$\aleph_0$> is.

You / Become a Mirror Of Me / When I Look In My Eyes / It Is You I See

All the properties we have just demonstrated are a manifestation of one common property of all infinite sets: reflexivity. When a set is reflexive, we can find a part of it that can be put into a bijection with the whole set. Learning a little about sets and subsets will allow us to express this property more precisely.

Subset Intermezzo

A subset is a part of set. The relation of “being a subset” is denoted by <$\subseteq$>. A few examples:

<$$ \displaylines{ \{1,2\} \subseteq \{1,2,3\} \\ \{d\} \subseteq \{a,b,c,d\} \\ \{2346,23531,132, 929875\} \subseteq \mathbb{N} \\ \{2,4,6,8,10,\ldots\} \subseteq \mathbb{N} } $$>

Note that an empty set <$\emptyset$> is a subset of every set. Also note that every set is a subset of itself.

<$$ \displaylines{ \emptyset \subseteq \{10,100,1000\} \\ \emptyset \subseteq \emptyset \\ \{x,y,z\} \subseteq \{x,y,z\} \\ \mathbb{N} \subseteq \mathbb{N} } $$>

A proper subset is a subset that doesn’t equal the whole set (it isn’t a subset of itself). Proper subsets are denoted <$\subsetneq$>.

<$$ \displaylines{ \{1,2,3,5\} \subsetneq \{1,2,3,4,5\} \\ \{95,96,97,98,99\} \subsetneq \mathbb{N} \\ \{1,2,3,\ldots,n\} \subsetneq \mathbb{N},\; \text{where } n \text{ is any finite natural number} } $$>

With this knowledge, we can restate reflexivity in mathematical terms: A reflexive set can be put into a bijection with a proper subset of itself.

Reflexivity obviously doesn’t make any sense for finite sets. You just can’t have a bunch of things, take one of them away and still have the same number of things. However, this common sense rule doesn’t apply to infinite sets. As we have seen, there are many operations that we can perform on infinite sets without breaking their sacred aura of infinitude.

What’s Next?

In the next article, we are going to take our minds for a thriller ride. Weren’t you wondering why there is that little zero in <$\aleph_0$>? Why would we need more than one Aleph? Surely there is just one infinity? Well, start preparing for a moderate shock, dear friends :-). We’ll see how we can mess with the sacred aura of infinitude…and blow it up a bit.

In the meantime, please give me feedback on this article. Was it interesting? Did it tell you anything new? Was it too easy? Too difficult? Too long? Too short? Did you miss the cute animals from previous parts? I welcome all constructive comments!

March 21, MMXI — Mathematics, Maths For Normal People, Infinity.