By Morse Anthony P.
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Extra info for A Theory of Sets
Logic 33 Definitional Axioms for Logic 33 Axioms of Definition for Logic 34 Axioms for Logic 34 xxix xxx Contents 2. Set Theory 41 Preliminaries 41 Orienting Definitions 41 Logical Definitional Axioms for Set Theory 43 Set-Theoretic Definitional Axioms for Set Theory 43 Axiom of Definition for Set Theory 43 Axioms for Set Theory 43 The Theorem of Extent 45 Some Aspects of Equality 49 Classification 5 1 The Theorem of Classification 52 The Role of Replacement 53 The Theorem of Replacement 54 The Theorem of Heredity 55 The Theorem of Subsets 55 The Theorem of Amalgamation 56 The Theorem of Unions 57 Singletons 57 Ordered Pairs 59 The Ordered Pair Theorems 63 Substitution 63 Unicity 66 The Theorem of Unicity 67 Relations 67 Functions 71 Ordinals 73 Definition by Induction 76 The General Induction Theorem 78 The Ordinary Induction Theorems 79 Choice 79 The Theorem of Choice 81 Maximality 8 1 Maximal Principle 83 Hausdorff’s Maximal Principle 85 The Inductive Principle of Inclusion 87 Well Ordering 88 The Well Ordering Theorems 89 Contents Natural Numbers 90 Sequences 91 Reiteration 93 Set Functions and Fixed Points 94 The Theorem of Bipartition 98 Equinumerosity 9% The Cantor-Bernstein Theorem 99 Cardinals 100 Cardinality 101 The Theorems of Cardinality 102 Cantor’s Power Theorem 104 Cardinal Arithmetic 105 Direct Extensions 106 Families of Sets 107 Tuples 110 A.
Our theory of notation and subsequent mathematical definitions will make possible a unique interpretation of the two parades just mentioned as well as a host of others. 34 AGREEMENT. A is of power n if and only if A is a nexus in which some symbol of type n appears and no symbol of type less than n appears. For example, are of power 6. 35 D E F I N I T I O N A L SCHEMA. We accept as a definition each expression which can be obtained by replacing ' A ' by an expression of odd power in any one of the expressions: '((X A X' A X " ) SZ ( ( X A X ' ) A X " ) ) ', ' ( ( X A X ' A X " A X m ) E ( ( X h X ' A X " ) AXm))', etc.
I n trying to make sure that our definitions conform to the Appendix we now pick up some loose ends. 65 DEFINITIONAL SCHEMA. We accept as a definition each expression which can be obtained from ‘(A;x=x)’ by replacing ‘A’ by a notarian not of class 5. 66 DEFINITIONAL SCHEMA. 67 =x)’ ‘ I ’ by a symbol of class 5. DEFINITIONAL SCHEMA. We accept as a definition each expression which can be obtained from ‘ (Ax ux = A x ux) ’ by replacing ‘A’ by an expression of either class 0 or class 1 or class 2. 68 DEFINITIONAL SCHEMA.
A Theory of Sets by Morse Anthony P.