Yi Lin. General Systems Theory: A Mathematical Approach

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Chapter 3

Finally, we assume γ is a limit ordinal and have defined the for each α < γ

satisfying (1)–(3) for α < β < γ. Let

and let

be a maximal disjoint subfamily of

Then (1) and (3) now hold for

all

α < β < γ. For β = γ, (2) says that no J ∈ is disjoint from all members

; this will follow by maximality of ιφ we can show that for each J∈

∃

K ∈

(

K ⊂ J). Let E be the set of all left and right endpoints of all intervals

E is countable and J is not separable, so fix K

∈ with K

⊂ J

and K

∩

If I ∈

∪

α<γ

α, thenK

does not contain the endpoints of I, so

∩

=or K

⊂ I. Now take K ∈ with K ⊂ K

and K

\cl(

K ) ≠ then

⊂

and

∈

(3.64)

Theorem 3.3.15. MA

(

) implies that there are no ω

-Suslin trees.

Proof:

Let (

, ≤) be an ω

-Suslin tree, and let P = (T, ≥), the reverse order

of T. Since T has no uncountable antichains, P has ccc. By Theorem 3.3.12, we

may assume that T is well-pruned, in which case D

= {

x ∈ T : ht(x, T) > α} is

dense in P. By

ΜΑ(

), there is a filter G intersecting each D

. Then G is an

uncountable chain, contradicting the fact that T is a Suslin tree.

First question: Is there anything which is not a set? We have declared that

our axioms of set theory deal with sets

— in fact, hereditary sets (Section 3.1).

Furthermore, the Axiom of Extensionality has embodied in it the assertion that all

things in our domain of discourse are sets. It seems likely that we have not left any

interesting mathematics behind by so restricting our universe, since mathematical

objects like the set of all real numbers, and the set of all complex numbers,

are hereditary sets and have been defined explicitly within this domain in Section

3.2.

Second question: Is there an “object” x such that x = {x}? This question is

independent of the existence of physical reality, and such a set x would clearly be an

hereditary set. Such an object x did not occur in the construction of mathematical

objects like

and

This is also an important question in systems theory (see

Chapters 8 and 12).

Some questions about sets are relevant to applications of mathematics. We

give two of the most important questions. A detailed discussion on the relevance

between the questions and applications of mathematics will be given in Chapter 8.

3.4. Well-Founded Sets

Axiomatic Set Theory

In this section, we study the position of the Axiom of Regularity relative to

the axioms in axiomatic set theory. We work in the axiom system of all axioms in

ZFC except Regularity and define the class WF of well-founded sets by starting

with and iterating the power set operation. Then we show that WF is closed

under the other set-theoretic operations as well.

Let us define R (

) for each

α ∈

by transfinite recursion as follows:

(a) R

(0) =

(b)

(c)

when

α is a limit ordinal.

A set A is well-founded if there is an

α ∈ ON such that A ∈ R(

). Let WF be the

class of all well-founded sets, i.e., WF =

∪

{

(

) : α ∈ ON

Theorem 3.4.1. For each ordinal α:

(a) R

(

) is transitive.

(b)

Proof:

We use transfinite induction on

: We assume that the result holds for

all

β < α and prove it for α.

Case 1.

α = 0. the result is trivial.

Case 2.

α is a limit. Statement (b) is immediate from the definition; (a)

follows from the fact that the union of transitive sets is transitive.

Case 3.

α = β +1. Since R

(

) is transitive, p

(

)) = R

(

) is transitive and

(

) ⊂ R

(

establishes (a) and (b) for all ordinals

Let x ∈ WF, the least ordinal α with x ∈ R(

) must be a successor ordinal.

The rank of

denoted by rank(

x), is the least β such that x ∈ R (β + 1). Hence,

β = rank(x), then x ⊂ R

(

and x ∉ R

(

), and x ∉ R

(

) for all α > β.

Theorem 3.4.2. For each ordinal

(

) = {

∈

: rank (

x ) < α

}

Proof:

For

∈

, rank

(

) < α if and only if ∃β < α

(

x ∈ R

(

β + 1)) if and

only if x

∈ R(

Chapter 3

Theorem 3.4.3. If y ∈ WF, then

(b) rank(

) =

∪

{rank(

x ) + 1 : x ∈ y}.

(a)

∀x ∈ y

(

x ∈ WF ∧ rank(x) < rank(y

)).

Proof: For (a), let α = rank(

), then y ∈ R (α + 1) = p (

(

)). If x ∈ y ,

∈

(

), so rank(x) < α.

For (b), let

α = ∪

{rank(

) + 1 : x

∈ y}. By (a) α ≤ rank(y). Furthermore,

each x

∈ y has rank < α, so y ⊂ R

(

α). Thus, y ∈ R

(

α +l), so rank(y) ≤ α.

Theorem 3.4.3 says that the class

is transitive and that we may think of the

elements

∈

as being constructed, by transfinite recursion, from well-founded

sets of smaller rank. Therefore, the class

excludes sets which are built up from

themselves. More formally, there is no x ∈ WF such that x ∈ x, since from this

we would have rank(x) < rank(x). Likewise, the class WF excludes circularities

∈

∧

∈

, since this would yield rank(x) < rank(y ) < rank(x). It is clear

that the following result holds.

Theorem 3.4.4. Each ordinal is in WF and its rank is itself.

Proof: What we need to show is that

∀α ∈ ON

(

α ∈

∧

rank(

α) = α ). We

prove this by transfinite induction on

α. Assume that the theorem holds for all

β < α. Then, for β < α, β ∈ R

(

+ 1) ⊂ R(

), so α ⊂ R

(

), so α ∈ R

(

+1).

By Theorem 3.4.3(b), rank(

) = ∪

{

β + 1 : β < α} = α. Thus the theorem holds

for all ordinals

α.

Theorem 3.4.5.

∀α ∈ ON

(

)

∩

The proof follows from Theorems 3.4.4 and 3.4.2.

Theorem 3.4.6.

(a)

If x

∈

, then x

y, x

∪

y, x

∩

{

}

, and y are all in WF, and the

rank of each of these sets is less than max{rank(x), rank(y)} +

If x

∈ WF, then ∪x, p

(

)

, and {x

}

∈

, and the rank of these sets is less

than rank(x) +

ω.

(b)

Proof: For (a), let α = rank(

). Then x ⊂ R

(

), so p (x) ⊂ p (R

(

)) =

(

α + l), so p

(

x) ∈ R

(

α + 2). Similarly, {x

}

∈

(

α + 2) and ∪x ∈ R(

+ 1).

For (b), let α = max{rank(x

), rank(

)}. As in (a), we show, e.g., {x, y}

∈

(

+2),(x

)

∈ R

(

α +3). Any ordered pair of elements of x ∪ y is in R

(

+ 3),

⊂

(

+ 3), so

∈

(

+ 4). We leave the rest of the details for the reader.

Axiomatic Set Theory 93

Theorem 3.4.7.

The sets

and

are all in R

(

ω + ω).

The proof follows from Theorem 3.4.6 and the definitions of these sets in

Section 3.2.

Theorem 3.4.8.

(a)

∀

∈ ω

)| < ω).

(b)

(

)|=

Proof:

(a) is easy by induction on

For (b), since

ω ⊂

(ω

), it is sufficient to

see that R

(

ω) is countable. To show this by induction, for each n we may identify

(

n + 1) with

(

)

2 and order it lexicographically. Then the ordered sum of the

sets R

(

), n ∈ ω, has order type ω.

The cardinalities of the sets R(

α) increase exponentially: |R(

)| = ω

(

+ 1)| = 2

, |R

(

ω + 2)| = 2

, etc. Generally, let a

be defined by trans-

finite recursion on

α as follows:

(1) a

= ω

(2) a

α +1

= 2

(3) For a limit

∪

{

: α < γ }.

Then we have the following theorem.

Theorem 3.4.9. |R(

+ 2)| = a

The proof follows from transfinite induction on

α.

The following theorem is an example of showing that all mathematics takes

place in the class WF.

Theorem 3.4.10.

(1)

Every group is isomorphic to a group in

(2) Every topological space is homeomorphic to a topological space in WF.

Proof: A group is an ordered pair (G,

× ) where × : G × G → G. Theorem

3.4.6 therefore implies that

(3.65)

If (

, × ) is any group, let α = |

| and let ƒ be a 1–1 mapping from α onto G.

Define an operation

on α by η = ƒ

–1

(

)

(

η)). Then (α

) is a group

Chapter 3

isomorphic to (

, × ). The same idea can be used to prove (b). We leave the proof

of (b) to the reader.

Generalizing the notion of well-orderings, we have the following: A relation

H is well-founded on a set A if and only if

(3.66)

The element y in Eq. (3.66) is called H -minimal in X. Thus, a relation H is

well-founded on A if and only if every nonempty subset of A has an H-minimal

element.

Theorem 3.4.11. If A

∈ WF, ∈ is well-founded on A.

Proof: Let X be a nonempty subset of A. Let

α = min{rank(y) : Y ∈ X }, and

pick a y

∈ X with rank(

) = α. Then y is ∈

-minimal in X by Theorem 3.4.3(a).

Theorem 3.4.12. If A is transitive and

∈ is well-founded on A, then A ∈ WF.

Proof: It is sufficient to show A

⊂ WF, because if A ⊂ WF, let α =

∪

{rank( y) + 1 : y ∈ x }; then x ∈ R (α), so x ∈ R(α + 1) ⊂ WF. If A ⊄ WF

let X = A

WF ≠ φ and let y be the ∈

-minimal in

. If z ∈ y, then z ∉ X

, but

∈

since A is transitive; so z ∈ WF . Thus, y ⊂ WF

, so

∈

. This contradicts the

definition that y

∈ A\ WF

Let A be a set. By recursion on n, we define

(1)

∪

A = A.

(2)

∪

+ 1

A= ∪

(

∪

(3)

tr cl(

A) = ∪

{

∪

A :

∈ ω

Here tr cl is the “transitive closure,” which is the least transitive set containing A

as a subset.

Theorem 3.4.13.

(a) A

⊂ tr cl

(

(b) tr cl(A

)

is transitive.

(d) If A is transitive, then tr cl(

) = A.

Axiomatic Set Theory

(e)

If x ∈ A, then tr cl(x

)

⊂

tr cl

(

(f)

tr cl(A) = A

{tr cl(

) :

∈

The proof is straightforward from the definition of transitive closure.

Theorem 3.4.14. For any set A the following are equivalent:

(a)

∈

(b)

tr cl(

)

∈

(c)

∈ is well-founded on tr cl(A

Proof: (a) → (b). If A ∈ WF, then by induction on n, ∪

A ∈ WF since WF

is closed under the union operation (Theorem 3.4.6). Thus, each

∪

⊂

, so

tr cl(A) ⊂ WF, so tr cl(A

)

∈

The proof of (b)

→

)

⊂

∈

Let V be the class defined by

(3.67)

In other words, V is the class of all sets. If we are convinced by the previous

theorems that all mathematics takes place in WF, it is reasonable to adopt as an

axiom the statement that V = WF. This does not mean that the two classes V and

are really identical, but only that we restrict our domain of discourse to be just

. In the following, we discuss some of the consequences of adding the axiom

V = WF to the axiom system of ZFC axioms except Regularity.

Theorem 3.4.15. The following are equivalent:

(a)

The axiom of regularity.

(b) ∀

(

∈ is well-founded on A).

Proof: (a)

↔ (b) is immediate from the definition of well-founded relations.

For (b)

→

(c), (b) implies that for any

∈

is well-founded on tr cl(

), so

∈

For (c)

→

(b), apply Theorem 3.4.11.

Unlike the other axioms of

ZFC, Regularity has no application in ordinary

mathematics, since accepting it is equivalent to restricting our attention to WF,

Chapter 3

where all mathematics takes place. In Chapters 5 and Chapter 9, we will see how

the Axiom of Regularity can be used to study level structures of general systems.

Since Regularity is equivalent to V = WF =

∪{R

(

) :

α ∈

}, it gives us a

picture of all sets being created by an iterative process, starting from nothing.

Theorem 3.4.16. A is an ordinal if and only if A is transitive and totally ordered

∈

Proof: To see that ∈ well-orders A, let X be any nonempty subset of A. Then

Regularity implies there exists an

∈-minimal element in X.

3.5. References for Further Study

There are many good books in the area of axiomatic set theory, such as (Drake,

1974; Engelking, 1975; Fraenkel et al., 1973; Jech, 1978; Kleene, 1952; Kreisel and

Krivine, 1967; Kunen, 1980; Kuratowski and Mostowski, 1976; Kurepa, 1936c;

Martin and Solovay, 1970; Quigley, 1970; Rudin, 1977; Shoenfield, 1975; Suslin,

1920).

CHAPTER 4

Centralizability and Tests of

Applications

The concept of centralized systems is written in the language of set theory in

order to take advantage of the rigorous mathematical reasoning. The concept of

centralizable systems is introduced. Applications of the concept in sociology,

concerning the existence of factions in human society, public issues of contention,

and importance level of problems versus media, are listed. Two real-life examples

are given to illustrate the results obtained in this chapter. At the end some open

questions are posed.

4.1. Introduction

Hall and Fagen (1956) introduced the concept of centralized systems, where a

centralized system is a system in which one object or a subsystem plays a dominant

role in the system operation. The leading part can be thought of as the center of

the system, since a small change in it would affect the entire system, causing

considerable changes. Lin (1988a) applied the concept of centralized systems

to the study of some phenomena in sociology. Several interesting results were

obtained, including the argument on why there must be a few people in each

community who dominate others. In this chapter we look at the overall study of

the concept of centralized systems, related concepts, and some applications.

n = n(

), a function of

, called the length of the relation r, such that r ⊆ M

In the rest of this section, all necessary concepts of systems theory and termi-

nology of set theory will be listed in order for the reader to follow the discussions.

A system is an ordered pair of sets, S = (M,R), such that M is the set of all

objects of S, and R is a set of some relations defined on M. The sets M and R

are called the object set and the relation set of S, respectively. Here, let r

∈ R

be a relation of S. Then r is defined as follows: There exists an ordinal number

Chapter 4

where

is the Cartesian product of n copies of M.

A system S = (

M,R

) is trivial (resp., discrete) if M = R = Ø (resp., M ≠ Ø

and either R = Ø or R = {Ø

}).

iven two systems S

= (

), i = 1,2,S

a partial system of S

if either (1) M

= M

and R

⊆ R

or (2) M

⊆

and

≠

there exists a subset R' ⊆ R

such that R

= R'

⏐

≡ {ƒ : ƒ is a relation on M

and there exists g ∈ R' such that ƒ is the restriction of g on M

}. In symbols,

⏐M

= {ƒ : g ∈ R'(ƒ = g⏐M

}, where

. The system S

is a

subsystem of S

if M

⊆ M

and for each relation

∈R

there exists a relation

∈

such that

⊆

⏐

A system S = (M,R

)

as n levels, where n is a fixed whole number, if (1)

each object S

= (M

) in M is a system, called the first-level object system;

(2) if is an (n – 1)th-level object system, then each object

is a system, called the n

th-level object system of

Let

= (

), i = 1,2, be two systems and h : M

→

a mapping. By

transfinite induction, two classes

, i =

1, 2, and a class mapping

can be defined with the properties

and for each x =

where Ord is the class of all ordinals. For each relation

is a relation on M

with length n(

). Without confusion, h will be used to indicate

the class mapping and h is a mapping from the system S

into the system

denoted by h : S

→ S

. When h : M

→ M

is surjective, injective, or bijective,

the mapping h : S

→

is also surjective, injective, or bijective, respectively.

The systems S

are similar if there exists a bijection h : S

→ S

such that

The mapping

is called a similarity mapping from

onto

. Evidently, if h is a similarity mapping from S

onto S

, the inverse

–1

is a similarity mapping from

onto S

. A mapping h : S

→

is termed a

homomorphism from

into S

, if h (R

) ⊆ R

Note that a theorem followed by (ZFC) means that the theorem is true only

under the assumption that the ZFC axioms are true, where ZFC stands for the

Zermelo–Fränkel system with the axiom of choice. For set X,

⏐X⏐ denotes the

Centralizability and Tests of Applications

cardinality of X. A cardinal number α = ℵ

is called regular if for any sequence of

ordinal numbers {α

ℵ

: β < λ

} , the limit of the sequence is less than

ℵ

. For

details and other terminology of set theory, see (Kuratowski and Mostowski, 1976).

4.2. Centralized Systems and Centralizability

To use the mathematical reasoning to study centralized systems, Lin

(submitted) redefined the concept in the language of set theory as follows.

Definition 4.2.1.

A system

= (

M,R

) is called a centralized system if each object

in S is a system and there exists a nontrivial system C = (M

) such that for

any distinct elements x and y

∈ M , say x = (M

) and y = (M

), then

= M

∩

and

the system C is called a center of S.

Theorem 4.2.1 [ZFC (Lin and Ma, 1993)]. Let

κ be an arbitrary infinite car-

dinality and

θ > κ a regular cardinality such that for any ordinal number

Assume that S = (

M,R

) is a system sat-

isfying

⏐

⏐ ≥ θ and each object m ∈ M is a system with m = (

) and

⏐M

⏐ < κ

If there exists an object contained in at least

objects in M, there then

exists a partial system S' =

(

M',R'

) of S such that S' forms a centralized system

and

⏐

⏐ ≥ θ

This result is a restatement of the well-known

∆

-lemma in axiomatic set theory

(Kunen, 1980). The following question, posed by Dr. Robert Beaudoin, is still

open.

Question 4.2.1. A system S = (

M,R

) is strongly centralized if each object in S is

a system and there is a nondiscrete system C =

(

, R

) such that for any distinct

elements x and y

∈ M, say x = (M

) and y = and

Give conditions under which a given system has a partial

system which is strongly centralized.

A system S

is n

-level homomorphic to a system

, where n is a fixed natural

number, if there exists a mapping h

→ A

, called an n

-level homomorphism,

satisfying the following:

(1) The systems

and A have no nonsystem kth-level objects, for each k < n.

(2)

For each object S

, there exists a homomorphism h

from the object

system

into the object system h

(

(3)

For each i < n and each ith-level object S

of S

, there exist level object

systems S

, for k = 0, 1, ... , i – 1, and homomorphisms h

, k = 1, 2, ... , i,

such that

is an object of the object system S

k –1

and h

is a

homomorphism from

into h

k–1

(

)

, for k = 1, 2, ... , i.