Prologue xi
Acknowledgments xix
Part I: The G¶delian Symphony 1
1 Foundations and Paradoxes 3
1 âThis sentence is falseâ 6
2 The Liar and G¶del 8
3 Language and metalanguage 10
4 The axiomatic method, or how to get the non-obvious out of the obvious 13
5 Peanoâs 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 Itâs raining paradoxes 30
14 Cantorâs diagonal argument 32
15 Self-reference and paradoxes 36
2 Hilbert 39
1 Strings of symbols 39
2 â⦠in mathematics there is no ignorabimusâ 42
3 G¶del on stage 46
4 Our first encounter with the Incompleteness Theorem ⦠47
5 ⦠and some provisos 51
3 G¶delization, or Say It with Numbers! 54
1 TNT 55
2 The arithmetical axioms of TNT and the âstandard modelâ N 57
3 The Fundamental Property of formal systems 61
4 The G¶del 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 Churchâs 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 G¶delian loop 84
6 âI Am Not Provableâ 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 Rosserâs proof 100
7 The Unprovability of Consistency and the âImmediate Consequencesâ of G1 and G2 102
1 G 2 102
2 Technical interlude 105
3 âImmediate consequencesâ 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 G¶delâs 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 G¶del 129
8 Bourgeois Mathematicians! The Postmodern Interpretations 131
1 What is postmodernism? 132
2 From G¶del to Lenin 133
3 Is âBiblical proofâ 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 âThe philosophical prejudices of our timesâ 151
4 From G¶del to Tarski 153
5 Human, too human 157
10 Mathematical Faith 162
1 âIâm not crazy!â 163
2 Qualified doubts 166
3 From Gentzen to the Dialectica interpretation 168
4 Mathematicians are people of faith 170
11 Mind versus Computer: G¶del and Artificial Intelligence 174
1 Is mind (just) a program? 174
2 âSeeing the truthâ and âgoing outside the systemâ 176
3 The basic mistake 179
4 In the haze of the transfinite 181
5 âKnow thyselfâ: Socrates and the inexhaustibility of mathematics 185
12 G¶del versus Wittgenstein and the Paraconsistent Interpretation 189
1 When geniuses meet ⦠190
2 The implausible Wittgenstein 191
3 âThere is no metamathematicsâ 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
