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