Logic, Induction and Sets

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Cambridge University Press, Jul 21, 2003 - Mathematics - 234 pages
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This is an introduction to logic and the axiomatization of set theory from a unique standpoint. Philosophical considerations, which are often ignored or treated casually, are here given careful consideration, and furthermore the author places the notion of inductively defined sets (recursive datatypes) at the center of his exposition resulting in a treatment of well established topics that is fresh and insightful. The presentation is engaging, but always great care is taken to illustrate difficult points. Understanding is also aided by the inclusion of many exercises. Little previous knowledge of logic is required of the reader, and only a background of standard undergraduate mathematics is assumed.
 

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Contents

Definitions and notations
5
11 Structures
6
12 Intension and extension
7
13 Notation for sets and relations
9
14 Order
11
15 Products
17
16 Logical connectives
20
Recursive datatypes
23
571 Further applications of ultraproducts
120
58 Exercises on compactness and ultraproducts
121
Computable functions
123
61 Primitive recursive functions
125
611 Primitive recursive predicates and relations
126
62 𝝁recursion
129
63 Machines
132
631 The 𝝁recursive functions are precisely those computed by register machines
134

213 Generalise from IN
24
214 Wellfounded induction
25
215 Sensitivity to set existence
39
217 Proofs
43
221 Propositional languages
47
223 Intersectionclosed properties and Horn formulae
49
224 Do all wellfounded structures arise from rectypes?
51
Partially ordered sets
52
312 Witts theorem
54
313 Exercises on fixed points
55
32 Continuity
56
321 Exercises on lattices and posets
60
33 Zorns lemma
61
331 Exercises on Zorns lemma
62
341 Filters
63
342 Atomic and atomless boolean algebras
66
35 Antimonotonic functions
67
36 Exercises
69
Prepositional calculus
70
41 Semantic and syntactic entailment
74
the Hilbert approach
75
412 No founders many rules
79
413 Sequent calculus
81
42 The completeness theorem
85
421 Lindenbaum algebras
89
423 Nonmonotonic reasoning
92
43 Exercises on propositional logic
93
Predicate calculus
94
52 The language of predicate logic
95
53 Formalising predicate logic
101
532 Predicate calculus in the natural deduction style
102
54 Semantics
103
541 Truth and satisfaction
104
55 Completeness of the predicate calculus
109
551 Applications of completeness
111
56 Back and forth
112
561 Exercises on backandforth constructions
114
57 Ultraproducts and Loss theorem
115
632 A universal register machine
135
64 The undecidablity of the halting problem
140
641 Rices theorem
141
65 Relative computability
143
Ordinals
147
71 Ordinals as a rectype
148
711 Operations on ordinals
149
712 Cantors normal form theorem
151
72 Ordinals from wellorderings
152
721 Cardinals pertaining to ordinals
159
722 Exercises
160
73 Rank
161
Set theory
167
81 Models of set theory
168
82 The paradoxes
170
83 Axioms for set theory with the axiom of foundation
173
84 Zermelo set theory
175
replacement collection and limitation of size
177
851 Mostowski
181
862 Von Neumann ordinals
182
863 Collection
183
864 Reflection
185
87 Some elementary cardinal arithmetic
189
871 Cardinal arithmetic with the axiom of choice
195
88 Independence proofs
197
881 Replacement
198
882 Power set
199
883 Independence of the axiom of infinity
200
885 Foundation
201
886 Choice
203
89 The axiom of choice
205
891 AC and constructive reasoning
206
892 The consistency of the axiom of choice?
207
Answers to selected questions
210
Bibliography
231
Index
233
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