Inaccessible cardinal: Difference between revisions
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(More specific example of aleph_1's regularity →Weakly inaccessible) |
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== Weakly inaccessible ==
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
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 in an ordinal collapsing function such as Madore's or Bachmann's.
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