Vipul: Created page with "==Definition== The '''cardinality of the continuum''' is a term used for an infinite cardinal defined in the following equivalent ways: # It is the cardinality of the se..."

2012-01-26T17:04:48Z

Created page with "==Definition== The '''cardinality of the continuum''' is a term used for an infinite cardinal defined in the following equivalent ways: # It is the cardinality of the se..."

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==Definition==

The '''cardinality of the continuum''' is a term used for an infinite [[cardinal]] defined in the following equivalent ways:

# It is the cardinality of the set of [[real numbers]].
# It is the [[power cardinal]] corresponding to the smallest infinite cardinal. Thus, it is the [[Beth number]] ''Beth one''.
# It is the cardinality of the [[closed unit interval]] <math>[0,1]</math>.
# It is the cardinality of any finite-dimensional [[Euclidean space]].
# It is the cardinality of any (finite-dimensional) [[manifold]].

===Assuming the continuum hypothesis===

The [[continuum hypothesis]] states that the cardinality of the continuum is the smallest uncountable cardinal, i.e., it equals <math>\aleph_1</math>. The continuum hypothesis is independent of ZFC (the standard axiomatic framework of set theory) but it follows from the [[axiom of constructibility]]. It is not generally considered to be either true or false.

Cardinality of the continuum - Revision history

Vipul: Created page with "==Definition== The '''cardinality of the continuum''' is a term used for an infinite cardinal defined in the following equivalent ways: # It is the cardinality of the se..."