Description: From canth2 , we know that ( aleph0 ) < ( 2 ^om ) , but we
cannot prove that ( 2 ^ om ) = ( aleph1 ) (this is the Continuum
Hypothesis), nor can we prove that it is less than any bound whatsoever
(i.e. the statement ( alephA ) < ( 2 ^om ) is consistent for any
ordinal A ). However, we can prove that ( 2 ^ om ) is not equal
to ( aleph_om ) , nor ( aleph( aleph_om ) ) , on
cofinality grounds, because by Konig's Theorem konigth (in the form of
cfpwsdom ), ( 2 ^om ) has uncountable cofinality, which eliminates
limit alephs like ( alephom ) . (The first limit aleph that is
not eliminated is ( aleph( aleph1 ) ) , which has cofinality
( aleph1 ) .) (Contributed by Mario Carneiro, 21-Mar-2013)