User blog comment:Simplicityaboveall/Simplicity/@comment-10429372-20141228090251

There is another mistake: Aleph_omega is not inaccessible, since its cofinalty is omega. The first inaccessible ordinal is much, much, much larger than, say, the omegath fixed point of alpha -> Aleph_alpha, or even the fixed point of that function. In fact, it is larger than everything you can come up with using the function alpha -> Aleph_alpha and recursion, just like Aleph_1 is larger than everything you can come up with using the function alpha -> omega^alpha and recursion.