The smallest infinity


When you think of the word infinity, what’s the first thing that comes to your mind? Perhaps the symbol of infinity (∞) or a vast amount of numbers just bombarded one after another indefinitely. In reality, infinity is not a number. It’s more of an idea; a concept that is yet to be well defined.  The concept of Aleph null or Aleph-nought brings together perhaps the two most extremely polar words in the English language. “Smallest” and “Infinity”. Yes, you guessed it. There are different levels of infinity with Aleph Null being the smallest of the lot.

Before we begin, Let’s take a quick peek at the function of cardinal numbers as an entity. Cardinal  Numbers are used as a standard of denoting how many things there are in a given area. For example, there are 5 bananas in the basket. Here 5 is the number used to denote the cardinality of (the amount of) bananas in the element. Similarly, if we take a set that entails the numbers 1, 2, 3, 4 and 5. We would say that the cardinality of the set is 5 i.e the number of elements in the set is 5. Easy enough? Now let’s fill up a new set with each and every natural number i.e every positive integer possible. The cardinality of this particular set is known as Aleph-Null and is denoted with the first letter of the Hebrew Alphabet ‘Aleph’ and the symbol (ℵo).

The interesting part of the Aleph null is that it isn’t merely limited to the total number of natural numbers. It also denotes how many even numbers there are, how many odd numbers there are and crazily enough, how many rational numbers( fractions) there are in the entirety of mathematics. Now, this statement might lead you to think that the even numbers and odd numbers would be the halves of all the natural numbers and that the rational numbers should be more than the natural numbers. Yet infinite sets behave differently compared to finite sets. Mathematician Georg Cantor discovered that two sets have the same number of elements if, and only if, those elements can be paired off. That is, there is a one-to-one correspondence between the two sets. He then further went on to prove how one can easily pair off the elements in the natural number set to the elements in the even/odd number set with a simple bijection(correspondence) of 2n↔️n. For example, the number 1 from the natural number set is paired to the first number of the even number set i.e 2, 2 can be paired off with 4 and so on till infinity. Here, every element of the natural number set will correspond to another element in the even number set and this occurrence is similar to the odd number set as well. Cantor’s proof shows that the total number of odd numbers/even numbers/rational numbers, etc are equal to the total number of natural numbers. Therefore with the bijection of 2n↔️n, Cantor similarly proved that the cardinality of the set of rational numbers also equals the smallest infinity, Aleph null.

Lastly, while one can correctly call ( ℵ0) the smallest infinity, the magnitude of the same is still incomprehensible and should not be underestimated. It is greater than any number one can think of. Despite being the smallest, it still bears the title of infinity.