The set has the same cardinality as the real numbers . This can be demonstrated by exhibiting surjective mappings from to the closed unit interval of and vice versa. Mapping onto is routine; map numbers less than or equal to (including ) to 0, numbers greater than or equal to (including ) to 1, and numbers between and to their equivalent in (mapping the infinitesimal neighbors of each dyadic fraction , along with itself, to ). To map onto , map the (open) central third (, ) of onto ; the central third (, ) of the upper third to ; and so forth. This maps a nonempty open interval of onto each element of , monotonically. The residue of consists of the Cantor set , each point of which is uniquely identified by a partition of the central-third intervals into left and right sets, corresponding precisely to a form in . This places the Cantor set in one-to-one correspondence with the set of surreal numbers with birthday .
the set has the same cardinality as the real numbers  this can be demonstrated by exhibiting surjective mappings from to the closed unit interval of and vice versa mapping onto is routine; map numbers less than or equal to (including ) to 0 numbers greater than or equal to (including ) to 1 and numbers between and to their equivalent in (mapping the infinitesimal neighbors of each dyadic fraction  along with itself to ) to map onto  map the (open) central third ( ) of onto ; the central third ( ) of the upper third to ; and so forth this maps a nonempty open interval of onto each element of  monotonically the residue of consists of the cantor set  each point of which is uniquely identified by a partition of the central-third intervals into left and right sets corresponding precisely to a form in  this places the cantor set in one-to-one correspondence with the set of surreal numbers with birthday 