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:

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'?
What is $\omega_1-\omega$ ? $\omega_2-\omega_1$ ? Or just any $\omega_n-\omega_{n-1}$ ?

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Aleph (ℵ\alephℵ) and Omega (ω\omegaω) Numbers Definition

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