[om-list] Cardinality Theory

[Mark Butler]

Here is my current representation of cardinality, if anyone is interested.

 - Mark Butler


/*
 * class Cardinality
 *
 * Cardinality - including transfinite numbers and various 
infinities       
 *
 * type     0 : natural / discrete, 1 : real / continuous
 * order    0 : finite, 1 : first infinity, 2 : second infinity, ...
 *                     -1 : first infinitesmal, -2 second infinitesmal, ...
 * value    value within order (must be normalized for orders > 0)
 *
 * Rules:
 *  1. under addition: type set to greatest,
 *     order adds if type equal, otherwise set to greatest
 *     value adds if type and order equal, otherwise set to value of
 *     higher (type,order)
 *     (lower (type,order) value is relative infinitesmal (effectively 
zero))
 *
 *  2. under subtraction: type set to greatest,
 *     order adds if type equal, otherwise set to greatest
 *     value subtracts if type and order equal, otherwise set to value of
 *     higher (type,order)
 *   
 *  3. under multiplication: type set to greatest, order adds, value 
multiplies
 *  4. under division: type set to greatest, order substracts, value divides
 *
 * Type, order examples:
 *
 * RO + R0 = R0   R0 - R0 = R0    R0 x R0 = R0    R0 / R0 = R0
 * R0 + R1 = R1   R0 - R1 = R1    R0 x R1 = R1    R1 / R0 = R1
 * R0 + R2 = R2   R2 - RO = R2    R0 x R2 = R2    R2 / R2 = R0
 *
 * N0 + R0 = R0   N0 - R0 = R0    N0 x R0 = R0    N0 / N0 = N0
 * N0 + R1 = R1   N0 - R1 = R1    N0 x R1 = R1    N1 / N0 = R1
 * N0 + R2 = R2   N0 - R2 = R2    N0 x R2 = R2    N2 / R2 = R0
 * 
 * N1 + N1 = N1   N1 - N1 = N1    N1 x N1 = N2    N2 / N1 = N1
 * N1 + N2 = N2   N1 - N2 = N2    N1 x N2 = N3    N3 / N1 = N2
 *
 * N1 + R1 = R1   N1 - R1 = R1    N1 x R1 = R2    R2 / N1 = R1
 * N2 + R1 = R2   N2 - R1 = R2    N2 x R1 = R2    R2 / R1 = R1
 * R1 + R1 = R1   R1 - R1 = R1    R1 x R1 = R2    R2 / R1 = R1
 * R1 + R2 = R1   R1 - R2 = R2    R1 x R2 = R3    R3 / R1 = R2
 *
 * Including coefficients:
 *
 * cardinality(all integers)          = N1
 * cardinality(all integers > 0)      = N1 / 2
 * cardinality(all even integers)     = N1 / 2
 * cardinality(semi-infinite line)    = R1 / 2
 * cardinality(infinite line)         = R1
 * cardinality(three infinite lines)  = 3 * R1
 * cardinality(quadrant)              = R2 / 4
 * cardinality(half plane)            = R2 / 2
 * cardinality(plane)                 = R2
 * cardinality(two planes)            = 2 * R2
 *
 * order(aleph0) = 1
 * order(aleph1) = 2
 */

class Cardinality
{
public:
 int         type;
 int         order;
 double      value;
};