Prologue xi
Acknowledgments xix
Part I: The Godelian Symphony 1
1 Foundations and Paradoxes 3
1 “This sentence is false” 6
2 The Liar and Godel 8
3 Language and metalanguage 10
4 The axiomatic method, or how to get the non-obvious out of the obvious 13
5 Peano’s axioms … 14
6 … and the unsatisfied logicists, Frege and Russell 15
7 Bits of set theory 17
8 The Abstraction Principle 20
9 Bytes of set theory 21
10 Properties, relations, functions, that is, sets again 22
11 Calculating, computing, enumerating, that is, the notion of algorithm 25
12 Taking numbers as sets of sets 29
13 It’s raining paradoxes 30
14 Cantor’s diagonal argument 32
15 Self-reference and paradoxes 36
2 Hilbert 39
1 Strings of symbols 39
2 “… in mathematics there is no ignorabimus” 42
3 Godel on stage 46
4 Our first encounter with the Incompleteness Theorem … 47
5 … and some provisos 51
3 Godelization, or Say It with Numbers! 54
1 TNT 55
2 The arithmetical axioms of TNT and the “standard model” N 57
3 The Fundamental Property of formal systems 61
4 The Godel numbering … 65
5 … and the arithmetization of syntax 69
4 Bits of Recursive Arithmetic … 71
1 Making algorithms precise 71
2 Bits of recursion theory 72
3 Church’s Thesis 76
4 The recursiveness of predicates, sets, properties, and relations 77
5 … And How It Is Represented in Typographical Number Theory 79
1 Introspection and representation 79
2 The representability of properties, relations, and functions … 81
3 … and the Godelian loop 84
6 “I Am Not Provable” 86
1 Proof pairs 86
2 The property of being a theorem of TNT (is not recursive!) 87
3 Arithmetizing substitution 89
4 How can a TNT sentence refer to itself? 90
5 γ 93
6 Fixed point 95
7 Consistency and omega-consistency 97
8 Proving G 1 98
9 Rosser’s proof 100
7 The Unprovability of Consistency and the “Immediate Consequences” of G1 and G2 102
1 G 2 102
2 Technical interlude 105
3 “Immediate consequences” of G1 and G 2 106
4 Undecidable 1 and undecidable 2 107
5 Essential incompleteness, or the syndicate of mathematicians 109
6 Robinson Arithmetic 111
7 How general are Godel’s results? 112
8 Bits of Turing machine 113
9 G1 and G2 in general 116
10 Unexpected fish in the formal net 118
11 Supernatural numbers 121
12 The culpability of the induction scheme 123
13 Bits of truth (not too much of it, though) 125
Part II: The World after Godel 129
8 Bourgeois Mathematicians! The Postmodern Interpretations 131
1 What is postmodernism? 132
2 From Godel to Lenin 133
3 Is “Biblical proof” decidable? 135
4 Speaking of the totality 137
5 Bourgeois teachers! 139
6 (Un)interesting bifurcations 141
9 A Footnote to Plato 146
1 Explorers in the realm of numbers 146
2 The essence of a life 148
3 “The philosophical prejudices of our times” 151
4 From Godel to Tarski 153
5 Human, too human 157
10 Mathematical Faith 162
1 “I’m not crazy!” 163
2 Qualified doubts 166
3 From Gentzen to the Dialectica interpretation 168
4 Mathematicians are people of faith 170
11 Mind versus Computer: Godel and Artificial Intelligence 174
1 Is mind (just) a program? 174
2 “Seeing the truth” and “going outside the system” 176
3 The basic mistake 179
4 In the haze of the transfinite 181
5 “Know thyself”: Socrates and the inexhaustibility of mathematics 185
12 Godel versus Wittgenstein and the Paraconsistent Interpretation 189
1 When geniuses meet … 190
2 The implausible Wittgenstein 191
3 “There is no metamathematics” 194
4 Proof and prose 196
5 The single argument 201
6 But how can arithmetic be inconsistent? 206
7 The costs and benefits of making Wittgenstein plausible 213
Epilogue 214
References 217
Index 225
