[om-list] Cardinality

[Mark Butler]

Cardinality

Instance cardinality of concepts and relations is a key analytical 
property.  For example, the concept father(s) has arbitrary instance 
cardinality, because any number of men can become fathers.  The relation 
father->son likewise has arbitrary cardinality.  However, John 
Adams->son does not.  John Adams had, at most, a small finite number of 
sons. Furthermore, the actual relation John Adams -> John Quincy Adams 
is of definite cardinality one, as are is the case of all  relations 
between physical entities.

In general if you have two concepts with possible instance cardinalities 
of M and N respectively, the maximum number of possible relations 
between discrete instances of those concepts is M x N.  But of course, 
for many abstractions (love, mercy, justice, etc.), there is no finite 
instance cardinality, nor even a countably infinite instance 
cardinality, but rather a continous infinite instance cardinality.  
Love, for example, is not countable, measurable perhaps, but definitely 
not countable.  The manifestations of love (such as light) may be 
quantitized, but love itself is ineffable.

Furthermore, from mathematical intuition, I am convinced that each 
Cantor countable infinity (aleph0, aleph1, ...) corresponds to one or 
more numbers.  Cantor cardinality is based on the concept of mapping 
members of one set to another. Any pair of sets that are mappable to 
eachother have the same Cantor cardinality.  However, the idea that 
Cantor cardinality is indistinguishable from number leads to physical 
absurdities, like the idea that a semi-infinite pole weighs the same as 
an infinite pole.

Elaborating on Cantor, I suggest the true cardinality (number) of a set 
has three aspects: Cardinality class (finite, countably infinity, 
continuous infinity),  cardinality sub class (e.g. aleph0, aleph1, 
....), and number within class and sub class (0,1,2,3,... for finite 
numbers, others speculative). 

Rather than modeling the properties of transfinite numbers, I have 
adopted symbols for the minimum and maximum number within each 
cardinality sub class.  That way I can clearly distinguish between the 
maximum finite cardinality and the minimum infinite cardinality - a 
distinction that is quite important in concept modelling.

For example, if I have a relation father->son, if I take both as plural 
the possible instance cardinality is countably infinite, because there 
are countably infinite numbers of possible fathers and possible sons. If 
I hold father singular, the possible instance cardinality is finite (at 
least in regards to mortality). 

The upper possible cardinality bound in each case is a key indicator 
that one is talking about the same concept in the same sense - something 
that is absolutely critical for comparative logic and the reason why all 
this talk of infinities is important for knowledge modeling.

Comments?

    - Mark


Example 16-bit cardinality representation:

enum /* cardinality classes */
 {
  CARDINALITY_FINITE               = 0x0000,
  CARDINALITY_COUNTABLE_INFINITY   = 0x1000,
  CARDINALITY_CONTINUOUS_INFINITY  = 0x2000,
  CARDINALITY_CLASS_MASK           = 0x7000
 };

enum /* cardinality sub-classes */
 {
  CARDINALITY_SUB_CLASS_MIN        = 0x0000,
  CARDINALITY_SUB_CLASS_ZERO       = 0x0000,
  CARDINALITY_SUB_CLASS_ONE        = 0x0100,
  CARDINALITY_SUB_CLASS_MAX        = 0x0F00,
 };

enum /* cardinality number (within class & subclass) */
 {
  CARDINALITY_NUMBER_MIN           = 0,
  CARDINALITY_NUMBER_MAX           = 0xFF,
  CARDINALITY_NUMBER_MASK          = 0xFF
 };
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