Symbols of infinity from Wikipaedia

In this post I will show an example of how mathematics can be very mysterious. I showed this to what might be classified as “first year in maths” students and they came out of class perplexed, as if their mind had to hit the reset button and reboot.

Theorem: Let $\mathbf{N}$ be the set of natural numbers and $\mathbf{Z}$ the set of integers. We have $\mathbf{N} \subseteq \mathbf{Z}$ and $|\mathbf{N}| = |\mathbf{Z}|$.

This means that though $\mathbf{N}$ is a subset of $\mathbf{Z}$, the size of $\mathbf{N}$ is equal to the size of $\mathbf{Z}$. In technical language the cardinality of the natural numbers is equal to the cardinality of the integers, What duh?

Proof:

a.) $\mathbf{N} \subseteq \mathbf{Z}$, this is trivial because the natural numbers $\mathbf{N}$ are just the positive integers in $\mathbf{Z}$. So every element of $\mathbf{N}$ is found in $\mathbf{Z}$.

b.) In order to show that the size of the two sets are equal we need to establish a bijective function from one set to the other. That is a function which is both surjective and injective. Another way of saying this is to say that we need a function that is onto and at the same time one-to-one from one set to the other. Obtaining such a function proves that the size of both sets are the same.

We will just get the one suggested by wikipaedia: We let
$f : \mathbf{Z} \rightarrow \mathbf{N}$ with $f(x) = 2|x|$ when and only when $x \in \mathbf{Z}$ and $x< 0$,

else $f(x) = 2x + 1$ when and only when $x \geq 0$.

1.) $f$ is one-to-one, i.e. injective. Let $f(x)= f(y)$, then we have two cases, either $2|x| = 2|y|$ or $2x+1 = 2y+1$. The first case we have $\Rightarrow |x| = |y|$, $\Rightarrow x,y < 0, \Rightarrow x = y$. On the other hand, if the second is the case then again this $\Rightarrow 2x+1=2y+1 \Rightarrow x = y$.

2.) $f$ is onto, i.e. surjective. Let $b \in \mathbf{N}$ then $b$ is positive and either even or odd. If even and positive, then in general $\Rightarrow b = 2|a|$ for some integer $a$,(the property of even numbers). Since $a$ is an integer, then it is an integer in $\mathbf{Z}$ and $|a| = b/2$ so that $f(a) = 2(b/2)=b$ and choose $a < 0$. On the other hand, if $b$ is positive and odd, then $b = 2a + 1$ for some integer $a$ (property of odd numbers) $\Rightarrow a = (b - 1)/2$ and $a$ has to be positive since $b$ is positive, i.e., $a \geq 0$ and so we have $f(a) = 2(b -1)/2+ 1 = b$. So in both cases we have seen that for every $b \in \mathbf{N}$ , we have found a matching $a \in \mathbf{Z}$. $\blacksquare$

How can a subset have the same size as it's superset? If this does not boggle your mind, perhaps you missed the point. The reason for this is that we have here two infinite sets, and this mystery only happens when infinity is involved. Now some mathematicians are not happy with this that is why they do not believe in infinite sets. It seems infinity is just a concept that has no matching physical reality and we can be indifferent with it. I suggest the concept of infinity is a metaphysical concept. So can one can reason that it is a concept that exists just in our mind and is not "real". Just like unicorns or fairies? I do not think so. There is no reason for us to believe in unicorns and in fact not all civilisations believe in the mythical horse. However the concept of infinity is different. It is because the concept per se is a necessity. The mind requires it when presented with the nature of numbers. For is it not true that the set of natural numbers is infinite? We can conceive it and by force of nature admit it. It is a necessary truth so in that sense infinite sets are real and transcends material physicality.