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Full Idea
An 'inaccessible' cardinal is one that cannot be reached by taking unions of small collections of smaller sets or by taking power sets.
Gist of Idea
An 'inaccessible' cardinal cannot be reached by union sets or power sets
Source
Penelope Maddy (Naturalism in Mathematics [1997], I.5)
Book Ref
Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.74
A Reaction
They were introduced by Hausdorff in 1908.