$\endgroup$ – pseudocydonia Jan 2 '19 at 15:40. Note that the above proves that $\aleph_0$ is a minimal element of the infinite cardinals. Aleph 1 is 2 to the power of aleph 0. The point is, there are more cardinals after aleph-null. MOAR INFINITY The infinity that you are probably talking about is the smallest infinity, called aleph-null. Repeated applications of power set will produce sets that can’t be put into one-to-one correspondence with the last, so it’s a great way to quickly produce bigger and bigger infinities. Since we define $\aleph_0$ to be the cardinality of $\Bbb N$, this means that every infinite subset of a set of size $\aleph_0$ is itself of size $\aleph_0$, and so there cannot be a smaller infinite cardinal. The smallest version of infinity is aleph 0 (or aleph zero) which is equal to the sum of all the integers. George Cantor proved alot of things about levels of infinity. This is called the cardinality of this smallest infinity. There infinity is Aleph one. Infinity means endless, but Trans-Infinity is bigger than Infinity!?. The number of irrational numbers is greater than the number of integers. TL;DR: In the same sense that there is no biggest natural number, there is no biggest infinity. The concept of infinity in mathematics allows for different types of infinity. It is also known as Aleph-null. Let’s try to reach them. i read from a book that aleph nought is greater than infinity but less than 2^(infinity) (i am not very sure about - so only this question). It’s an infinity bigger than aleph-null. There is no smaller. Those two sets have the same number of members because you can put them into 1-1 correspondence. They can not be put into a 1-1 correspondence. That infinity is called Aleph null. There is no mathematical concept of the largest infinite number. pl give more details and reference websites I recently was looking up facts about different cardinalities of infinity for a book idea, when I found a post made ... {\aleph_\omega}$ is bigger than $\aleph_\omega$. If you want, you can add one to it, and the cardinality wouldn’t change.