Yi Lin. General Systems Theory: A Mathematical Approach
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Index
(G) integrals, 332
(
m, n
) -derivative, 321
1–1 mapping, 16
Abstract distances, 344
AC, 62
Addition, 27
Addition of order types, 35
Algebraic geometry, 344
Almost disjoint family, 78
Ancient atomists, 114
Antichain, 33
Apollo project, 5
Arithmetic of natural numbers, 168
Associated input–output system, 299
Atom, 113
Atomic theory, 114
Attractor, 258
Attributes, 9
Axiom of choice, 62
Axiomatic set theory, 59
Beauty of mathematics, 164
Bellman’s Principle of Optimality, 137, 148
Bellman’s principle of optimality, 135
Bernstein’s Equipollence Theorem, 25
Big sum, 328
Bijective mapping, 16
Black box, 2
Boolean algebra, 344
Box product topology, 325
Burali-Forti paradox, 57
Calorists, 114
Cartesian product of two sets, 22
Catastrophe theory, 10
Causal relation, 5
Causal system, 276
Chain, 33
Chain of object systems, 116, 123
Chaos, 12, 256
Class of all sets, 95
Classical systems theory, 10
Cofinal, 34
Cofinality, 75
Coinitial, 34
Color, 164
Compartment theory, 11
Complement, 19
Complex system representation, 292
Complexity, 6
Component system, 281
Concatenation, 274
Conservation of matter, 113
Conservation of mechanical energy, 115
Consistent set, 61
Continuous everywhere, 325
Continuously ordered set, 41
Continuum hypothesis, 75
Crystalline material, 174
Crystallization process, 175
Cut, 41
Cybernetics, 2
Data analysis approach, 8
De Morgan’s Laws, 20
Decision theory, 2
Decomposability, 309
Decoupled by feedback, 285
Dedekind cut, 345
Densely ordered set, 40
Density, 336
Denumerable set, 16
Derivative, 325
379
380
Index
Descartes’ second principle, 3
Dimension theory, 344
Dirac’s
δ function, 124
Dirac’s quantum mechanics, 124
Distribution, 125
Divisibility, 111
Dynamic programming, 135
Einstein’s relativity, 124
Elementary functions, 345
Elementary space, 337
Empty set, 19
Energies, 5
Epimorphism, 294
Epistemology, 174
Equipment-replacement models, 135
Equipollence, 16
Exponentiation, 27
Extensionality, 62
Exterior measure, 326
Factor system, 281
Feedback component, 266
Feedback decoupling, 309
Feedback invariant property, 271
Feedback system, 265, 309
Feedback transformation, 266
Filter, 80
Finite set, 73
Finitist, 64
First-order predicate calculus, 59
Formal language, 171
Formal power series, 344
Formalist, 64
Formula, 60
Foundations of geometry, 344
Four-element philosopher, 113
Function, 16, 67
Functional system, 265
Fundamental equation, 136, 143
Fundamental particle, 111
Fundamental Theorem of Calculus, 346
Fuzzy mathematics, 11
Galileo’s method, 4
Game Theory, 12
Game theory, 2
General linear system, 309
General system, 123
General systems methodology, 6
Generalized continuum hypothesis, 75
Generalized number, 126, 311
Generalized number system, 125
Generalized power series, 125
Generalized Principle of Optimality, 139
Generalized solution, 326
Genetic algorithm, 12
Global system representation, 292
GNL-integrable function, 133
GNL-integral, 330
GNS function induced by f, 341
Gödel completeness theorem, 64
Gödel’s theorem, 172
Graph of a function, 327
Graph theory, 12
Heat of friction, 114
Heaviside function, 345
Height, 84
Hierarchy of monads, 3
Hu’s counterexample, 136
Human activity systems approach, 7
Hyperreal number, 188
Identical systems, 123
Impassable chasm, 180
Increasing function, 43
Indefinite integral, 335
Infinite set, 73
Infinitesimal, 125, 314
Infinities, 125
Infinity, 314
Information highway, 5
Injective complex system representation, 302
Injective mapping, 16
Input space, 255
Input–output system, 255
Input–output system covering, 298
Input–output system partition, 298
Interior measure, 327
Inventory problem, 135
Isolated ordinal, 70
Isomorphism, 67, 295
i
th-order generalized extreme value, 157
κ
-disjoint sets, 77
Index
381
Lavoisier’s Axiom, 117
Law of Conservation of Fundamental Particles,
117
Lebesgue integral, 328
Lebesgue integration, 325
Lebesgue measurable set, 327
Length of a relation, 123
Level index, 126, 311
Limit of the λ
-sequence, 48
Limit ordinal, 47
Linear order relation, 33
Linear ordering, 67
Linear system, 263
Linear time system, 274
Linear uniformity, 344
Linearly independent functions, 339
Mapping, 16, 67
Martin’s Axiom, 80
Materials science, 174
Mathematical induction, 179
Mathematical proofs, 181
Maximal, 33
Maximum, 33
Medieval mysticism, 3
Minimal, 33
Minimum, 33
Modeling relation, 293
Molecular spatial structure, 174
Monad, 188
Monomorphism, 296
MT-time system, 273
Multilevel structure, 122
Multiplication, 27
Music, 166
Naive set theory, 15
Navier–Stokes equation, 12
Network, 13
Newton’s law of action and reaction, 115
Non-Archimedean analysis, 344
Non-Archimedean number field, 125
Non-Archimedean topology, 344
Noninteracted system, 281
Nonstandard analysis, 125
n
th-level object system, 116, 123
nth-order optimization solution, 157
Objects, 9
ω
µ
-additive topological spaces, 344
One-element Ionians, 112
One-to-one mapping, 16
Open interval, 326
Operation Epitome Principle, 149
Optimization model, 150
Optimization model, 148
Optimum-preserving law, 138
Order topology, 87
Order type, 35
Ordered algebraic structures, 344
Ordered pair, 10, 22
Ordered set, 32
Ordered sum, 35
Ordinal exponentiation, 72
Ordinal number, 45
Ordinalities, 45
Ordinals, 45
Original system, 266
Output space, 255
Overall system, 281
p
-adic topology, 344
p
-resolving mapping, 155
Painting, 166
Pansystems theory, 309
Paradox of a moving particle, 180
Paradox of the hotel, 180
Partial function, 23
Partial order, 80
Participant approach, 7
Peano’s axioms, 168
Platonists, 64
Poetry, 166
Power set, 26, 62
Precausal system, 276
Problem-oriented systems approaches, 6
Product mapping, 293
Product of order types, 36
Products of distributions, 346
Proper subset, 19
Quantifier, 60
Quantum field theory, 345
Quantum mechanics, 346
Quasi-(
m
,
n
) continuity, 317
Quasi-(
m
,
n
) limit, 321
Regular ordinal, 75
Regularity, 62
Relation, 10, 67
Resource allocation problems, 135
Rest mass, 124
Riemann integral, 326
Russell’s paradox, 57, 123
382
Index
Schwartz’s distributions, 346
Science of science, 1
Scientific revolution, 4
Seeds, 113
Self-controlled equipment, 5
Sentence, 61
Set, 15
Set existence, 62
Set theory, 10
Shift operator, 274
Shortest-path problem, 135
Similarity mapping, 34
Simple function, 327
Simulation, 13
Small sum, 328
Stationary set, 148
Statistics, 13
Stone duality theorem, 344
Strange attractor, 260
Strict total ordering, 67
Strictly order-preserving function, 149
Strictly strongly antisymmetric relation, 150
Strongly antisymmetric relation, 150
Strongly stationary system, 275
Structured systems analysis approach, 8
Subformula, 60
Subsystem, 123
Successor, 70
Successor ordinal, 70
Surjective complex system representation, 302
Surjective mapping, 16
Suslin line, 88
Suslin tree, 85
Suslin’s hypothesis, 88
System with relations, 9
Systemality, 6
Systems connected in series, 291
Systems description of mathematics, 172
Systems movement, 347
Systems research, 4
Systems theory, 2
Taylor expansion, 345
Test function, 336
Theory of automata, 14
Theory of distributions, 335
Theory of science of science, 176
Three-body problem, 5
Time invariably realizable system, 277
Topological space, 87
Total ordering, 67
Totality, 6
Traditional approach, 7
Transfer principle, 345
Transfinite Induction, 72
Transfinite Recursion, 72
Traveling-salesman problem, 135
Tree, 84
∆
-system, 78
∆
-System Lemma, 79
Type class, 38
Unbordered set, 40
Uncountable set, 16
Uniform convergency, 326
Unit cell, 174
Universal closure, 61
Utopian project, 4
Vase puzzle, 178
Volume of a interval, 326
Voronese’s non-Archimedean geometry, 344
Weak derivative, 325
Well-founded set, 91
Well-ordered set, 42
Well-Ordering Theorem, 52
White box, 2
Wholeness, 1
Zermelo–Russell paradox, 57
Zero, 138
Zeroth-order optimization solution, 156
ZFC axiom system, 59