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

Aleph-Null (0\aleph_0) is the cardinality of the set of the natural numbers and Omega (ω\omega) is the infinite ordinal corresponding to that cardinal. But 0+1\aleph_0+1 still equals 0\aleph_0 while ω+1\omega+1 does not equal ω\omega! I get this when I read about Hilbert's Hotel. Same with that 20=02\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 xixi, where xx is a real number and ii is the imaginary unit) and complex numbers is 1\aleph_1 and the ordinal that corresponds to it is ω1\omega_1! The Aleph and Omega numbers do continue on.

My Questions:

  1. What is the definition of 2\aleph_2 or 3\aleph_3 or 4\aleph_4 or just any n\aleph_n for that matter other than 'It's the next biggest infinite cardinal'?
  2. What is ω1ω\omega_1-\omega? ω2ω1\omega_2-\omega_1? Or just any ωnωn1\omega_n-\omega_{n-1}?

Note by Lâm Lê
2 weeks, 2 days ago

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I read that there are three of these ‘alephs’,and 2\aleph _2 stands for the number of all geometric curves.

Jeff Giff - 2 weeks, 1 day ago

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

Jeff Giff - 2 weeks, 1 day ago

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