[om-list] Numbers as Operators

[Mark Butler]

Numbers as Operators

In standard number theory, a number is thought to be equivalent to the 
set of all sets that have the specified number of members, i.e. those 
that have the specified cardinality.  However, in natural language, 
numbers are not sets, but rather operators on sets.

A number operating on a set is similar in operation to a power set 
operator operating on a set.   A power set is the set of all the subsets 
of a set. If you have a set with N elements, there are 2^N distinct 
subsets.  A number operating on a set produces a set containing all the 
subsets of a set.  So if I have the set "presidents of the United 
States", and operate on it with the number "3",  I get "three presidents 
of the United States", or rather, one of the subsets of original set 
that contain three members, {Washington Adams Jefferson}, {Truman 
Kennedy Reagan}, or whatever.

In English, classes are always singular, and sets of unknown cardinality 
are always plural.  You can convert any class to a set simply by 
changing it from singular to plural.  "apple" is a class. "apples" is a 
set.    Singular definite expressions are always treated as classes, not 
sets.   "my apple" and "the apple" are classes with one instance.  But 
"my apples" and "the apples" are sets.

We say "Give me one of the apples" - "one of" is a set membership 
operator and "apples" is a set.  Alternatively, when we say "Give me one 
apple", "apple" is a class and "one" is a singular class subset 
operator, one that converts a class to a set (as necessary) and then 
extracts one member, resulting in the same semantics as the first statement.

In English, it is not natural to have sets of sets, only sets of 
classes.  Sets in English always combine via union ("A, B, C and D, F, 
G") if left to their own devices.  If you want to do the equivalent of 
sets of sets you have to convert to classes first, indeed any 
conventional mathematical set is technically an English class because it 
is treated as a singular expression.  So "A, B, C" is an English set, 
but {A, B, C} is an English class.  A mathematical set of sets (e.g. 
"{{A,B},{C}}")  leads to Russell's Paradox when applied to the set of 
all sets that do not contain themselves.

However, the same syntax read in  English, as "the class of all classes 
that are not subclasses of themselves" leads to no paradox, becauses no 
class can ever contain a proper super class of itself, by definition - 
no recursion, no paradox.  That is an unfair change in semantics, but it 
is nice to know that even though it implements an unusually 
sophisticated arbitrary order logic, English is not easily prone to the 
paradoxes that (regrettably) leveled Frege's set theoretic system.

- Mark



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