A Computational Approach to the Ordinal Numbers Documents ordCalc 0.3.2 Paul Budnik Mountain Math Software [email protected] Copyright c 2009 - 2012 Mountain Math Software All Rights Reserved Licensed for use and distribution under GNU General Public License Version 2. Source code and documentation can be downloaded at www.mtnmath.com/ord and sourceforge.net/projects/ord. Contents List of Figures 5 List of Tables 5 1 Introduction 7 1.1 Intended audience . 8 1.2 The Ordinals . 8 2 Ordinal notations 9 2.1 Ordinal functions and fixed points . 10 2.2 Beyond the recursive ordinals . 11 2.3 Uncountable ordinals . 12 3 Generalizing Kleene's O 13 3.1 Objective mathematics . 13 3.1.1 Avoiding the ambiguity of the uncountable . 14 CK 3.1.2 Objective mathematics beyond !1 ................... 15 3.2 Kleene's O ..................................... 15 3.3 P an extension of Kleene's O .......................... 16 3.3.1 P conventions . 17 3.3.2 P definition . 17 3.3.3 Summary of rules for P ......................... 19 1 3.4 Q an extension of O and P ........................... 20 3.4.1 Hierarchies of ordinal notations . 20 3.4.2 Q syntax . 21 3.4.3 Q conventions . 22 3.4.4 Ranking labels (lb) in Q ......................... 24 3.4.5 Q definition . 25 3.4.6 Summary of rules for Q ......................... 27 3.5 Conclusions . 28 4 Ordinal Calculator Overview 28 4.1 Program structure and interactive mode . 30 4.1.1 Program structure . 30 4.1.2 Interactive mode . 31 4.2 Recursive ordinal notations and the Cantor normal form . 31 4.3 The Veblen hierarchy . 32 4.3.1 Two parameter Veblen function . 32 4.3.2 Finite parameter Veblen functions . 33 4.3.3 Transfinite Veblen functions . 36 4.4 Limitations of ordinal notations . 36 4.5 Notations for the Church-Kleene ordinal and beyond . 38 4.5.1 Notations for countable admissible ordinals . 39 4.5.2 Admissible ordinals and projection . 41 4.6 Ordinal projection with nested embedding . 44 4.7 Mathematical truth . 47 4.7.1 Properties of the integers . 54 4.7.2 The reals . 55 4.7.3 Expanding mathematics . 56 5 Program structure 57 5.1 virtual functions and subclasses . 57 5.2 Ordinal normal forms . 58 5.3 Memory management . 58 6 Ordinal base class 58 6.1 normalForm and texNormalForm member functions . 59 6.2 compare member function . 60 6.3 limitElement member function . 60 6.4 Operators . 60 6.4.1 Addition . 62 6.4.2 multiplication . 62 6.4.3 exponentiation . 63 2 7 The Veblen hierarchy 63 7.1 The delta operator . 65 7.2 A finite function hierarchy . 67 7.3 The finite function normal form . 68 7.4 limitElement for finite functions . 68 7.5 An iterative functional hierarchy . 70 8 FiniteFuncOrdinal class 70 8.1 compare member function . 71 8.2 limitElement member function . 74 8.3 fixedPoint member function . 77 8.4 operators . 77 8.4.1 multiplication . 77 8.4.2 exponentiation . 80 8.5 limitOrd member function . 82 9 IterFuncOrdinal class 82 9.1 compare member function . 83 9.2 limitElement member function . 84 9.3 fixedPoint member function . 87 9.4 operators . 87 10 Countable admissible ordinals 90 10.1 Generalizing recursive ordinal notations . 90 10.2 Notations for Admissible level ordinals . 91 10.3 Typed parameters and limitOrd ........................ 92 10.4 limitType of admissible level notations . 95 10.5 Ordinal collapsing . 95 10.6 Displaying Ordinals in Ψ format . 99 10.7 Admissible level ordinal collapsing . 99 11 AdmisLevOrdinal class 102 11.1 compare member function . 102 11.2 limitElement member function . 107 11.3 isValidLimitOrdParam member function . 119 11.4 limitInfo, limitType and embedType member functions . 119 11.5 maxLimitType member function . 119 11.6 limitOrd member function . 119 11.7 fixedPoint member function . 123 11.8 Operators . 125 12 Nested Embedding 125 12.1 Filling the Gaps . 126 3 13 NestedEmbedOrdinal class 133 13.1 compare member function . 133 13.2 class NestedEmbeddings ............................ 136 13.2.1 compare member function . 136 13.2.2 nextLeast member function . 136 13.3 limitElement member function . 136 13.4 isValidLimitOrdParam and maxLimitType member functions . 147 13.5 limitInfo, limitType and embedType member functions . 147 13.6 limitOrd member function . 147 13.7 fixedPoint member function . 147 14 Philosophical Issues 147 A Formalizing objective mathematics 151 A.1 Introduction . 151 A.1.1 The mathematics of recursive processes . 151 A.1.2 The uncountable . 151 A.1.3 Expanding the foundations of mathematics . 152 A.2 Background . 152 A.2.1 The ordinal hierarchy . 153 A.3 The true power set . 153 A.4 Mathematical Objects . 154 A.4.1 Properties of properties . 155 A.4.2 G¨odeland mathematical creativity . 155 A.4.3 Cantor's incompleteness proof . 155 A.5 Axioms of ZFC . 155 A.5.1 Axiom of extensionality . 156 A.5.2 Axiom of the empty set . 156 A.5.3 Axiom of unordered pairs . 156 A.5.4 Axiom of union . 156 A.5.5 Axiom of infinity . 157 A.5.6 Axiom scheme of replacement . 157 A.5.7 Power set axiom . 158 A.5.8 Axiom of Choice . 158 A.6 The axioms of ZFC summary . 158 A.7 The Objective Parts of ZF . 159 A.8 Formalization of the Objective Parts of ZF . 160 A.8.1 Axioms unchanged from ZF . 160 A.8.2 Objective axiom of replacement . 160 A.9 An Objective Interpretation of ZFC . 161 A.10 A Creative Philosophy of Mathematics . 161 4 B Command line interface 163 B.1 Introduction . 163 B.2 Command line.