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." Ph.D. thesis. Göttingen, Germany, 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).
