The Schröder-Bernstein theorem for numbers states that if
then
For sets, the theorem states that if there are injections
of the set into the set and of into , then there is a bijective correspondence
between
and
(i.e., they are equipollent).
Bernstein, F. "Untersuchungen aus der Mengenlehre." PhD dissertation. Göttingen, Germany: University of Göttingen, 1901.Bernstein,
F. "Untersuchungen aus der Mengenlehre." Math. Ann.61, 117-155,
1905.Schröder, E. "Ueber zwei Definitionen der Endlichkeit
und G. Cantor'sche Sätze." Nova Acta Academiae Caesareae Leopoldino-Carolinae
(Halle a.d. Saale)71, 303-362, 1898.Schröder, E. "Die
selbständige Definition der Mächtigkeiten 0, 1, 2, 3 und die explicite
Gleichzahligkeitsbedingung." Nova Acta Academiae Caesareae Leopoldino-Carolinae
(Halle a.d. Saale)71, 365-376, 1898.
For sets, the theorem states that if there are injections
of the set 
into the set 
and of 
into 
, then there is a bijective correspondence
between 
and 
(i.e., they are equipollent).
