This is called the cardinality of this smallest infinity. 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 $\aleph_\omega$. The concept of infinity in mathematics allows for different types of infinity. Aleph 1 is 2 to the power of aleph 0. The number of irrational numbers is greater than the number of integers. Infinity means endless, but Trans-Infinity is bigger than Infinity!?. That infinity is called Aleph null. It’s an infinity bigger than aleph-null. pl give more details and reference websites 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). $\endgroup$ – pseudocydonia Jan 2 '19 at 15:40. Those two sets have the same number of members because you can put them into 1-1 correspondence. George Cantor proved alot of things about levels of infinity. 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. The point is, there are more cardinals after aleph-null. They can not be put into a 1-1 correspondence. Let’s try to reach them. MOAR INFINITY The infinity that you are probably talking about is the smallest infinity, called aleph-null. It is also known as Aleph-null. TL;DR: In the same sense that there is no biggest natural number, there is no biggest infinity. There is no mathematical concept of the largest infinite number. Note that the above proves that $\aleph_0$ is a minimal element of the infinite cardinals. 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. If you want, you can add one to it, and the cardinality wouldn’t change. 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