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== Weakly inaccessible ==
Weakly inaccessible cardinals were first invented.introduced Essentially,by cardinalsHausdorff suchin asan \(attempt \aleph_\omegato \)resolve arethe called[[continuum limithypothesis]].<ref>Hausdorff, cardinalsFelix. because"[https://gdz.sub.uni-goettingen.de/id/PPN235181684_0065?tify={%22pages%22:%5b453%5d} theyGrundzüge can'teiner beTheorie reachedder fromgeordneten belowMengen]", viaMathematische finiteAnnalen, iterationsvol. of65, thenum. successor4 operation: if \(1908), \kappapp.435--505. DOI:10.1007/BF01451165.</ref> Cardinals \(\aleph_\omega alpha}\), thenfor \(limit \kappa <ordinal \aleph_n (\alpha\) forare someknown \(as nlimit \)cardinals, thussince \(applying the cardinal successor operator to a cardinal less than \kappa^{+(m\aleph_\alpha\)} <yields \aleph_{na +cardinal m}also <less than \(\aleph_\omega alpha\). (citation for this being the etymology? {{citation needed}}) However, \( \aleph_\omega \) and many other limit cardinals have a short cofinal sequence, -these thus theycardinals are called singular. Formally, \( \kappa \) is singular if there is some sequence of \( < \kappa \) smaller ordinals whose limit is \( \kappa \). For example, \( \aleph_\omega \) is the limit of the sequence \( \aleph_0 \), \( \aleph_1 \), \( \aleph_2 \), ... which has length \( < \aleph_\omega \). Meanwhile, for cardinals such as \( \aleph_1 \), arethe unreachablelimit fromof belowany viacountable mechanismssequence suchof ascountable transfiniteordinals recursionis countable, andso theno limitsystem of anynormal countablefunctions can build up a sequence ofshorter countablethan ordinals\( is\aleph_1 countable\) cofinal in \( \aleph_1 \) - thus \( \aleph_1 \) is not singular - aka regular. We now call a cardinal \( \kappa \) weakly inaccessible if it is regular ''and'' a limit cardinal. You can see that if \( \kappa \) is weakly inaccessible and \( \alpha < \kappa \) then \( \aleph_\alpha < \kappa \) too, and then regularity gives that the limit of the length-\( \omega \) sequence \( \alpha \), \( \aleph_\alpha \), \( \aleph_{\aleph_\alpha} \), ... is less than \( \kappa \) as well.
The existence of weakly inaccessible cardinals is - surprisingly - not provable in \( \mathrm{ZFC} \), assuming its existence. We explain why in the third section. However, authors may typically assume their existence and use them in ordinal collapsing functions to describe ordinals equal to or greater than the [[Extended Buchholz ordinal|EBO]], since an inaccessible cardinal acts as a suitable "diagonalizer" over \( \alpha \mapsto \Omega_\alpha \), like how \( \Omega \) acts as a suitable diagonalizer overin an ordinal arithmeticcollapsing function such as Madore's or Bachmann's.
== Strongly inaccessible ==
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