section_path: Cardinality/Infinite sets and infinite cardinality
section_name: Infinite sets and infinite cardinality
paragraph_text: Infinite sets and infinite cardinality
  
  The list of elements of some sets is endless, or infinite. For example, the set N {\displaystyle \mathbb {N} } of natural numbers is infinite. In fact, all the special sets of numbers mentioned in the section above are infinite. Infinite sets have infinite cardinality.
  
  Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of N {\displaystyle \mathbb {N} } are called countable sets; these are either finite sets or countably infinite sets (sets of the same cardinality as N {\displaystyle \mathbb {N} }); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of N {\displaystyle \mathbb {N} } are called uncountable sets.
  
  However, it can be shown that the cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.