Theory of formal systems

157 Pages·1996·7.113 MB·English

by  Smullyan, Raymond Merrill

THEORY OF
FORMAL SYSTEMS
BY
Raymond M. Smullyan
PRINCETON, NEW JERSEY
PRINCETON UNIVERSITY PRESS
Copyright © 1961, by
Princeton University Press
All RightsReserved
L. C. Card60-14063
This research was mainly supported jointly by the Department of
the Army,theDepartment oftheNavy,and theDepartment ofthe
AirForce underAir Force ContractNo. AF 19(604)-5200. It was
alsosupportedinpartbya NationalScience Foundationgrant-in-
aidin KnotTheoryand Metamathematics, Mathematics Depart­
ment,PrincetonUniversity.
Fifth printing, 1996
TABLE OF CONTENTS
Page
PREFACE............................................... vii
CHAPTER I: FORMAL MATHEMATICAL SYSTEMS.................... 1
#A. Elementary Formal Systems...................... 2
§0. Motivation.............................. 2
§i . Definition of an Elementary Formal System..... 3
§2. Alternative Formulation of Elementary
Formal Systems......................... 5
§3- Representability......................... 6
§1. Mathematical Systems...................... 9
#B. Recursive Enumerability.......................... 10
§5- Recursively Enumerable Attributes of Positive
Integers............................... 10
§6. Godel Numbering.......................... 11
§7. The Universal System U................... 12
§8. The Recursive Unsolvability of U........... 14
Appendix................................. 16
CHAPTER II: FORMAL REPRESENTABILITY AND RECURSIVE ENUMERABILITY. 19
§o. Some Preliminary Principles................ 19
#A* Closure Properties.............................. 20
§1 . Closure Under Existential Definability....... 21
§2. Solvability over K....................... 26
#B- Recursive Enumerability.......................... 28
§3- Recursive Enumerability of Some Basic
Arithmetical Attributes................... 28
§4. Recursive and Partial Recursive Functions.... 29
§5- Finite Quantification; Constructive Definability 29
#C. Transformations on Alphabets; Godel Numbering........ 31
§6. Extension of Alphabets................. 31
§7- I)yadic Godel Numbering................. 32
§8. Solvability.......................... 33
§9- Lexicographical Ordering; n-adic Representation
of Numbers............................. 34
§io. Admissible Godel Correspondences.............. 37
§11. Further Facts About Admissibility [Optional].... 37
#D. A Brief Summary................................ 38
CHAPTER III: INCOMPLETENESS AND UNDECIDABILITY............... 39
#A. Incompleteness................................. 4i
§1 . Representation Systems................. 41
§2. First DiagonallzatIon Lemma; Tarski’s Theorem... 42
§3. Consistency and Completeness; Godel’s Theorem... 45
§4. Complete Representability and Definability In Z. 46
§5- Separability within Z; Rosser’s Theorems.. 47
§6. Symmetric Systems.......................... 49
§7- Extensions........................... 50
#B. Undecidability................................. 51
§8. Systems With an "Effective" Representation
Function............................... 52
CONTENTS
Page
§9* Undecidability......................... 53
§10. Normality.............................. 55
§11. Additional Theorems..................... 55
§12. Universal Systems....................... 56
§13- Undecidability and Incompleteness.......... 56
#C. Undecidability and Recursive Inseparability...... 58
§14. Definability in Formal Systems............ 58
§15. Extensions............................. 58
§16. Recursive Inseparability................. 58
§17- Separation of R. I Sets WithinS ystems...... 59
§18. Rosser Systems.......... .............. 60
§19* Recursive Inseparability of the Diagonal
Sets T*, R*......................... 61
CHAPTER IV: RECURSIVE FUNCTION THEORY..................... 65
#A. Effective Operations and Fixed Point Theorems.... 65
§1. Enumeration Theorem..................... 65
§2. Indexing.............................. 67
§3. Iteration Theorems for r.e. Relations...... 67
§4. Effective Operations................ .... 70
§5* Fixed Point Theorems.................... 72
§6. Double Recursion Theorems................ 75
#B. Constructive Arithmetic and Rudimentary Attributes.. 77
§7. Some Preliminaries...................... 78
§8. Dyadic Concatenation.................... 79
§9« Rudimentary Attributes................... 81
§10. Pure Elementary Formal Systems............ 84
§11. Arithmetization of Elementary dyadic
Arithmetics......................... 85
#C. Enumeration and Normal Form Theorems............ 89
§12. Kleene Enumeration Theorem............... 89
§13. Separation of Differences of r.e. Sets..... 90
§14. Partial Recursive Functions.............. 90
§15. Functional Indexing..................... 91
CHAPTER V: CREATIVITY AND EFFECTIVE INSEPARABILITY.......... 93
#A. Creativity and Effective Inseparability......... 93
§1. Productive and Creative Sets; Recursive and
Effective Inseparability............... 93
§2. Many-one and One-one Reducibility......... 95
§3. Creative Systems....................... 97
§4. Effective Inseparability................. 98
§5. Effective Rosser Systems................. 99
#B. Further Theory of Productive Sets............. 100
§6. Weakly Productive Functions.............. 100
§7. Uniform Reducibility.................... 102
§8. Universal Sets......................... 104
§9. On Uniform Reducibility.................. 104
§10. Uniform Representability................. 106
§11. Bl-Unlformity.......................... 106
#C. Effective Inseparability and Double Productivity.... 107
§12. Doubly Productive Pairs................. 107
§13- Reducibility of Pairs to Pairs............ 110
#D. Double Universality.......................... 112
§14. Doubly Universal and Totally Double Universal
Pairs.............................. 112
§15. Uniform Reducibility ................ 112
CONTENTS
Page
§16. Weakly Doubly Productive Pairs............ 113
§17- On Doubly Productive Pairs................ 11 4
§18. Application to Rosser Systems............. 117
§19- Uniform Reducibility..................... 117
§20. A Generalization of Effective Inseparability... 119
§2i. Total Double Universality................. 120
#E. Double Isomorphisms.......................... 123
§22. Double Isomorphism...................... 123
§23. 1 - 1 Equivalence and Double Isomorphism..... 125
§24. Double Isomorphism of Rosser Systems........ 126
SUPPLEMENT........................................... 127
§1. Theories............................... 127
§2. Calculability of Functions; Normality....... 130
§3. ^-Consistency; Enumerability Within (T);
Godel1 s Theorem....................... 131
§4. Rosser*s Construction.................... 132
§5« Godel Theories and Rosser Theories......... 134
§6. Theories in Which Plus and Times are Definable. 136
§7- Some Special Theories.................... 137
§8. Essential Undecidability................. 137
§9* Essential Creativity..................... 138
§10. Exact Rosser Theories.................... 139
REFERENCE AND BRIEF BIBLIOGRAPHY"

PREFACE
This study combines an introduction to recursive function theory
(and its applications to metamathematics) with a presentation of new
results in the field. The author has particularly borne in mind the needs
of the generally mature mathematician with no background in mathematical
logic. Our treatment (particularly in Chapters I and II) has been mainly
influenced by the elegant methods of Post.
Chapter I commences with a new characterization of "formal math-
matical systems" and "recursively enumerable sets and relations". We in­
troduce the notion of an "elementary formal system" which serves as the
basic formalism for the entire study. A very short and simple proof is
given of Church’s theorem — that there exists no uniform "algorithm" for
deciding which sentences are provable in which mathematical systems. The
proof is in the spirit of Post, but the normal form theorem for canonical
systems is avoided. The study of elementary formal systems is continued
in Chapter II, which consists mainly of results of a preliminary nature
for the remaining chapters.
In Chapter III we approach the Gddel and Rosser incompleteness
theorems and related results on undecidability, from a highly abstract
point of view. The usual machinery of mathematical logic (the prepositional
and first order functional calculi) is not employed. The applications to
mathematical logic proper are treated separately in the supplement. The
results on undecidability are all deduced from a tiny fragment of recursive
function theory developed in #A of Chapter II. Chapter III also extends
some well known metamathematical results; these are further extended in
Chapter V.
Chapter IV contains a connected presentation of recursive function
theory from a viewpoint which combines the theory of elementary formal sys­
tems with an extension of Quine’s techniques of concatenation theory. [The
reader whose main interest is in recursive functions can read this chapter
directly following Chapter II.] GSdel’s program of arithmetizing syntax
is accomplished in a new manner; no appeal is made to primitive recursive
vii
function theory, prime factorization, theory of congruences or the Chinese
remainder theorem. A by-product of this approach (which was undertaken
primarily out of considerations of elegance) is that improved normal form
theorems are obtained.
The concluding chapter contains the results of the author’s
recent research on the theory of universal sets and double universal pairs.
A particularly interesting application, jointly due to Hilary Putnam and
the author, is given in the concluding section of the supplement.
This study is a revision of the author’s recent doctoral
dissertation [31 ]• The author wishes to express his thanks to Alonzo
Church, John Myhill and Hilary Putnam for their kind encouragement and
help. Thanks are also due to Mrs. Euthie Anthony, for her competent
secretarial help, and to Robert Ritchey and Robert Windor for help in proof
reading. Special thanks are due to James Guard, who undertook the final
reading of the entire manuscript, and supplied numerous corrections and
suggestions.
Raymond M - Smullyan
Princeton, New Jersey
August, 1960
viii
the memory of my father