In one of the letters written to the Infeld group in Warsaw Einstein wrote:
"A new manner of thinking is essential if humankind is to survive."

  
     MODELING COURSE B21: Introduction to Set Theory. 

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MODELING COURSE B21: Introduction to Set Theory
NOTE By "ref" we shall mean below:
http://evans-experientialism.freewebspace.com/metanomskiindex.htm
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UNDERLYING LOGIC
----------------
Each axiomatic theory is Founded in some Founding or Underlying theory
whose Axioms and Theorems it accepts as granted, as its own "extrinsic"
Axioms. The underlying discipline of the Set Theory is Logic.
Which Logic precisely?
Here we run against a problem.  

At the end of the 19th century two breaches appeared in the apparently 
stable and consistent pyramid of science:
-Michelson's experiment showing invariance of light speed which 
 shattered the bases of Physics,
-Russell's Paradox which shattered the bases of Logic.

Unlike Einstein who decided to rebuild breached Physics from the 
scratch, logicians preferred local patches of the breach which resulted 
in the crisis of Logic staying open till today and in numerous not 
entirely consistent, competing logical systems. Best specialists of the 
Set Theory disagree about the underlying Logic opting for "Functional 
Calculus with Equality", for "Predicate Calculus", for "Propositional 
Calculus with well formed formulas", etc.

In this situation we felt compelled to define our own logical system 
which we presented in ref. "Crisis of Logic" and which endeavors to 
satisfy the post-Carnapian requirements (ref. "The Post Carnapian 
Situation"): 

After Carnap, Logic has to satisfy following requirements:
-Fuzzy, "swampy", gray "probability" or "certainty" replacing the 
 traditional black and white "truth/falsity",
-Axiomatic ultimate premises similar to Popper's "piers" driven into 
 "swamps",
-Factual ultimate conclusions similar to physical observations,
-Unlimited dimensionality supporting deduction from any number of 
 premises and induction from any number of conclusions, which implies 
 network as fundamental logical structure.
-Customized logic: algorithms of fuzzy operators vary from application 
 to application.

We shall consider henceforth this system as the underlying Logic of 
the Set Theory and call it shortly "PC" (for Propositional Calculus).

The present course is limited to Exact Set Theory supported by the 
Exact Proposition Calculus as defined in ref. "The Post Carnapian 
Situation" in which the Certainty takes only two values 1 and 0 
symbolizing "true" and "false".

Fuzzy Set Theory founded in Fuzzy Logic will be discussed in a future 
course dedicated to Artificial Intelligence.

Definitions, Symbols and Conventions:
-------------------------------------
"Attribute" and "Entity" as defined in Modeling Course A considered
to be meaningful, i.e. understandable as defined in ref. Model of
Mind (Lateral thinking).
 
"Proposition" or "Statement" will be understood as any assignment of
   a meaningful Attribute to a meaningful Entity, i.e. any English 
   meaningful sentence where by "meaningful" we mean additionally
   "logically determinate", i.e. whose Certainty may be set via valid
   inference rules of PC. 

"Predicate" is the assignment operator of the Proposition.

In PC we have defined 16 two dimensional Operators and shown how they
map into higher dimensionality. Set Theory uses overwhelmingly five of 
them which we shall symbolize with mixed upper/lower case character 
chains:

o14: And (intersection)                    (1000)
o4:  Or  (union)                           (1110)
o7:  Orr, (either or), (disjunction)       (0110)
o2:  Imp (implication) (If)                (1011)
o8:  Eq (equivalence) (Iff, If and ony If) (1001) 

They map one to one from two to n dimensions with exception of Orr
which forks into "One of", "Two of", etc. We shall consider Orr as
"One of" which is consistent with its function in 2D.

Further we have defined in PC the unary operator "negation" which
we shall symbolize with "Not".

In order to stay homogeneous with the usual terminology of the Set 
Theory we introduce the term "Formula" symbolized with "Frm(x)" 
synonym of "Proposition" in the above defined sense, whose subject 
is "x".

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NAIVE SET THEORY (NST)
----------------------
The "Naive" Set Theory adds to the underlying PC the term "Collection" 
(of Entities) taken as granted intuitively, without rigorous 
definition. 

For NST "Set" is any rigorously investigated Collection and thus
every Collection is a potential Set. "Set" means simply a Collection 
investigated by the Set Theory and structured with help of Set
Theoretical Axioms. We shall see below two basic ones, the Axiom of 
Extensionality and the Axiom of Comprehension.
(Collections and Sets are  symbolized with lower case character chains 
(x, azer, horses)).     

NST adds to the operators of underlying Logic four Set Theoretical
operators;

The "Contained" Predicate "Cnt": 
   x Cnt y reads "(Entity) x is contained by (Set) y", 
   or "y contains x", or "x is Member (or Element) of y".

The Universal Qualifier "All":
   All x reads "for all x". All x (x Cnt y) reads "for all x contained
   by y", or "for all x, Members of y".

The Existential Qualifier "Ex":
   Ex x reads "there exists an x". Ex x (x Cnt y) reads "there exist
   an x contained by y".

The "Included" Predicate "Incl":
   x Incl y reads "(Set) x is included in (Set) y", or "x is Subset 
   of y". y may be called informally "Superset of x".

Definition of Subset:
All z ((z Cnt x) Eq (z Cnt y)) Imp x Incl y
In words: If all Members of x are Members of y then x is Subset of y.

Corollary: Each set is Subset of itself.

Note: Beginners have often difficulty to distinguish Member from 
Subset. Let's illustrate their difference with an example:
-Any "animal" is Member of the Set "animals". 
-Set "horses" is Subset of "animals" as every "horse" is "animal".
-Set "horses" is not an "animal", thus not Member of "animals".
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Axiom of Extensionality
-----------------------
All x ((x Cnt y) Eq (x Cnt z)) Imp y Eq z 
In words: If y and z have the same Members they are equivalent, or
the logically true converse: equivalent Sets have the same Members.

This Axiom had a particularly strong importance for the NST, but seen
from today's perspective it's one of many examples showing how the 
operations of underlying Logic can be broadened over Sets with 
help of the additional Set-Theoretical operators. For instance logical Or 
can be broadened over Sets as follows:

All x, All y ((x Cnt z) Eq (y Cnt z)) Imp (z Eq (x Or y)) 
In words Iff all Members of x and all Members of y are Members of z
then z is Union of x and y. 

With the Axiom of Extensionality and its implications we have secured
broadening of underlying logical operations over Sets so that we can
deal with existing Sets. However, we don't know yet if and how new 
Sets can be constructed. The second basic Axiom takes care of that:

Axiom of Comprehension
----------------------
Ex y, All x ((x Cnt y) Eq Frm(x)) [AC]
In words: All x satisfying Frm(x) are contained in the (new) Set y.

So we can both, construct Sets and deal with them; all seems best in
the best possible world, in the "Cantorian Paradise". But a closer
look at the tree of knowledge expels us from the Paradise: putting
Frm(x) = "x Not Cnt x" we get Russell's Paradox saying that not all
Collections are Sets:

Indeed:
Ex y, All x ((x Cnt y) Eq (x Not Cnt x))
encompasses all instances of x thus also the instance "x=y":

Ex x All x ((x Cnt x) Eq (x Not Cnt x)) [RP symbolic]

which is the famous Russell's Paradox, in words, For "x", the Set of 
all Sets not being Members of themselves:

x is Member of itself iff x is not Member of itself [RP in words]

Thus, "x", the Set of all Sets not being Members of themselves is a
contradiction and consequently does not exist. Symbolically:
Not Ex x All x ((x Cnt x) Eq (x Not Cnt x)) [NRP]

Yet the Collection of Sets not being Members of themselves obviously 
exists and contains the overwhelming majority of Sets such as  Set of 
horses which is not a horse, Set of numbers which is not a number, etc. 

In x we have discovered a Collection which is not a Set, which 
falsifies the basic claim of NST, namely that all Collections are Sets
thus refuting the NST and calling for some more general Theory capable 
to deal rigorously with Set- and Non-Set Collections. As we shall see 
below, current Set Theories indeed endeavor to deal rigorously with 
Non-Set Collections which they call "Classes".

Let's note that [NRP] refuting an instance of [AC] refutes entirely 
the Axiom so that instead of 
Ex y, All x ((x Cnt y) Eq Frm(x)) 
we must assert:
Not Ex y, All x ((x Cnt y) Eq Frm(x))
which leaves the NST without means of constructing new Sets. 

In the following paragraph we shall see how current Set Theories solve
this problem dealing rigorously with Non-Set Collections or "Classes".
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CURRENT SET THEORY
------------------
One can say without exaggeration that the overwhelming part of research 
concerning the Set Theory expelled from the naive Cantorian Paradise 
endeavors to overcome Russell's and other paradoxes and to create 
a paradox-free Set Theory. Due to the persisting Crisis of Logic this 
research resulted in numerous competing theories. From the point of 
view of designer of Information Systems most if not all of them share  
a new concept of "Class".

"Class" symbolized with upper case character strings (X, AZER, HORSES)
   denotes a Collection which is not a Set or, more precisely, about 
   which we cannot say if it's a Set, but which may eventually become 
   one in the light of some subsequent evidence.

There is only one difference between Class and Set: unlike Set, Class
may not be Member of any Collection, i.e. may not be Contained.
It may only be Included in Sets or Classes as "Subclass".

Applying the concept of Class to the undetermined Collection y in the
refuted Axiom of Comprehension [AC] we get the "extended" Axiom of 
Comprehension:

Ex Y, All x ((x Cnt Y) Eq Frm(x))   [EAC]

Checking [EAC]'s instance "Frm(x) = x Not Cnt x" we get the solution 
of Russell's Paradox:

Ex Y, All x ((x Cnt Y) Eq (x Not Cnt x )  [SRP]
in words: There exists a CLASS Y Containing all Sets which are not 
Members of themselves.

[SRP] is consistent and free of contradiction indicating that [EAC]
supplies a pertinent construction rule. However, this rule applies
only to Classes and in order to be able to construct Sets we still
need some rule of conversion of Classes to Sets. Various Set 
Theories deal with this problem in different usually rather complex
ways. For our needs it's sufficient to say that In order to construct 
a Set "y" we have to:

1.Construct the class "Y" with help of [EAC] for a particular Formula
  Frm(x)=Phi(x): Ex Y, All x ((x Cnt Y) Eq Phi(x)) [EAC(Phi)] .

2.Substitute Set "y" for "Y" getting 
  Ex y, All x ((x Cnt y) Eq Phi(x)) [AC(Phi)]

3.Prove explicitly or implicitly that all instances of [AC(Phi)]
  are consistent. 

For Information System Designer 1,2,3 mean:

1.Defining ER Model in terms of Virtual Entities or Classes and 
  their Relations and implementing it as "Schema" with help of some 
  chosen software.

2.Populating Schema with test data.

3.Testing.
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