I’ve always had in my head the question: what is more “infinite”, the infinity of natural numbers or the infinity of integer numbers?
The concept of infinity is both magical and dangerous, it’s confused, we have heard and used this concept since basic education without caution, simply we have used it.
If you ask somebody, everybody will tell you that integer numbers are infinite, that there are infinity numbers, people use the word “infinity” when they are talking about numbers, I wanna go ahead, what’s the meaning of “there are infinity numbers”?
Maybe sometimes everybody have asked himself the question I started this post with, there are more integer numbers than natural numbers?, or clearly, is the set of integer numbers bigger than the set of natural numbers? Maybe the question sounds stupid, it could be true, What a nonsense, numbers are infinite! or natural numbers are a type of integer numbers, obviously there are more integer numbers than naturals!
The first answer sounds offensive, while second is interesting and reasonable , but to find a better answer, we have to study a little of sets theory, cause we are talking about sets of numbers, like are N and Z.
When we talk about N and Z, we refer to the sets of natural and integer numbers. Like every set, they are compound of elements:
The number of elements of a set is the cardinality of that set, |N| or |Z|. We have more concepts about sets, now we have to know what is the cartesian product of two sets and the concept of function derived of him.
The cartesian product of two sets A and B, A x B is the set formed by all the ordered pairs such as (a,b) where a€A and b€B. The position of a and b is significant,so A x B /= B x A. Example:
A function from A to B is a subset of AxB where an element of A only appears at the first position of a unique par (an element of set A can’t be in more than one pair). A is the domain of the function and B is the range. Example:
Another example with numbers:
In the last example, for each element of the domain (natural numbers) there is only one element of the range that belongs to him (his square).
In a 1 to 1 function, each element of the range appears only one in the set f. Example:
With these concepts, we just can try solve our question. First, we can check if |N|=|Z|. The equality is true if can find a function 1 to 1 of N over Z, that means, find a function 1 to 1 such as his range be all the set Z, but after many tries of searching that function, we noticed that does not exists, we can’t find a range of f with all the elements of the set Z with a 1 to 1 function with domain in N.
We can check if |Z|=|N| ,seems that our question could be solved finding the absolute value function:
This function verifies condition of function Z over N, but isn’t a function 1 to 1, because for every element of range N, there are two elements of domain Z with image on it. Bad.
At this point, we can say that |N|/=|Z|, we should check if |N|<|Z|. A easy function like this could help us:
We have found a function 1 to 1 of N to Z, so we can say that |N|<|Z|. Finally we have demostrated that set of integer numbers is more “infinite” than the set of natural numbers.
— Sure, I’m bigger than you, little set of natural numbers — said Z to N.