  then κ is said to be (strongly)inaccessible. Since  ℵ0 is inaccessible, it is normal practice to confine attention to those inaccessible, or weakly inaccessible, cardinals that exceed  ℵ0. Accordingly, inaccessible or weakly inaccessible cardinals will always be taken to be uncountable. It is clear that such cardinals—if they exist—must be extremely large; and indeed the Gödel incompleteness theorem implies that the existence of even weakly inaccessible cardinals cannot be proved from the usual axioms of set theory. 