Define alpha_0 = 0, alpha_n+1 = aleph_alpha_n. Let alpha = sup{alpha_n : n is a natural number). Prove that alpha = aleph_alpha.(adsbygoogle = window.adsbygoogle || []).push({});

My attempt: As alpha <= aleph_alpha is obvious, I've been trying to prove the other direction of inequality, so that being both <= and >= implies =, but now I'm not even sure if this is the right approach. I think I cannot use (transfinite) induction because this isn't a statement about n, so I've been stuck with

sup{alpha_n : n is a natural number) >= sup{aleph_beta : beta < alpha}

where the RHS is just the definition of a cardinal aleph_gamma where gamma is a limit ordinal. Maybe I can find an injection from the RHS to the LHS but it doesn't seem to work either. Any help will be appreciated.

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# Prove that alpha = aleph_alpha where

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