Inaccessible cardinal: Difference between revisions

Weakly inaccessible cardinals were first introduced by Hausdorff in an attempt to resolve the [[continuum hypothesis]].<ref>Hausdorff, Felix. "[https://gdz.sub.uni-goettingen.de/id/PPN235181684_0065?tify={%22pages%22:%5b453%5d} Grundzüge einer Theorie der geordneten Mengen]", Mathematische Annalen, vol. 65, num. 4 (1908), pp.435--505. DOI:10.1007/BF01451165.</ref> Cardinals \( \aleph_\alpha \) for limit ordinal \( \alpha \) are known as limit cardinals<ref>Jech, Thomas (2003), ''Set Theory'', Springer Monographs in Mathematics (third millennium ed.), Berlin, New York: Springer-Verlag</ref>, since applying the cardinal successor operator to a cardinal less than \( \aleph_\alpha \) yields a cardinal also less than \( \aleph_\alpha \). However, \( \aleph_\omega \) and many other limit cardinals have a short cofinal sequence, these cardinals are called [[Cofinality|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 \( \aleph_1 \), the limit of any countable sequence of countable ordinals is countable, so \( \aleph_1 \) 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.