# Aleph ($\aleph$) and Omega ($\omega$) Numbers Definition

Aleph-Null ($\aleph_0$) is the cardinality of the set of the natural numbers and Omega ($\omega$) is the infinite ordinal corresponding to that cardinal. But $\aleph_0+1$ still equals $\aleph_0$ while $\omega+1$ does not equal $\omega$! I get this when I read about Hilbert's Hotel. Same with that $2\aleph_0=\aleph_0$ so the cardinality of the set of integers is the same as the set of natural numbers. It is also the same as the set of rational numbers (see An easy proof that rational numbers are countable)! However the cardinality of the set of irrational and real and imaginary (probably, because a pure imaginary number is just $xi$, where $x$ is a real number and $i$ is the imaginary unit) and complex numbers is $\aleph_1$ and the ordinal that corresponds to it is $\omega_1$! The Aleph and Omega numbers do continue on.

My Questions:

1. What is the definition of $\aleph_2$ or $\aleph_3$ or $\aleph_4$ or just any $\aleph_n$ for that matter other than 'It's the next biggest infinite cardinal'?
2. What is $\omega_1-\omega$? $\omega_2-\omega_1$? Or just any $\omega_n-\omega_{n-1}$?

Note by Lâm Lê
8 months, 1 week ago

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I read that there are three of these ‘alephs’,and $\aleph _2$ stands for the number of all geometric curves.

- 8 months, 1 week ago

Also note that the number of complex numbers is equal to the number of dots on a plane.

- 8 months, 1 week ago

I remember a video I watched a while ago that said the ω's continue, each one infinitely bigger than the last. Same goes for the ℵ's. That means one minus the last is what you started with.

- 7 months ago