Georg Cantor proved via diagonalization that this set has a cardinality of (*) aleph-one, which is the cardinality of the power set of the integers; that result implies a bijection between this set and the open interval from zero to one.
A field that is "formally [this adjective]" can be equipped with an ordering to become an ordered field. The only complete Archimedean ordered field is comprised of these numbers. These numbers can be constructed  from  another set by Dedekind cuts because  that other  set is dense in the numbers described by this adjective. Georg Cantor proved via diagonalization that this set has a cardinality of (*) aleph-one, which is the cardinality of the power set of the integers; that result implies a bijection between this set and the open interval from zero to one. According to the fundamental theorem of algebra, these numbers lack algebraic closure because polynomials with coefficients of either complex numbers or these numbers have complex roots. For 10 points, name these numbers that comprise the number line.
A field that is "formally [this adjective]" can be equipped with an ordering to become an ordered field.
 real  [or  reals ; or formally  real  field; or  real  numbers; or  R ]