There's Something About G¶del

Acknowledgments.

Part I: The G¶delian Symphony.

1 Foundations and Paradoxes.

1 "This sentence is false".

2 The Liar and G¶del.

3 Language and metalanguage.

4 The axiomatic method, or how to get the non-obvious out of the obvious.

5 Peano's axioms ¢¦ .

6 ¢¦ and the unsatisfied logicists, Frege and Russell.

7 Bits of set theory.

8 The Abstraction Principle.

9 Bytes of set theory.

10 Properties, relations, functions, that is, sets again.

11 Calculating, computing, enumerating, that is, the notion of algorithm.

12 Taking numbers as sets of sets.

13 It's raining paradoxes.

14 Cantor's diagonal argument.

15 Self-reference and paradoxes.

2 Hilbert.

1 Strings of symbols.

2 "¢¦ in mathematics there is no ignorabimus".

3 G¶del on stage.

4 Our first encounter with the Incompleteness Theorem ¢¦ .

5 ¢¦ and some provisos.

3 G¶delization, or Say It with Numbers!

1 TNT.

2 The arithmetical axioms of TNT and the "standard model" N.

3 The Fundamental Property of formal systems.

4 The G¶del numbering ¢¦ .

5 ¢¦ and the arithmetization of syntax.

4 Bits of Recursive Arithmetic ¢¦ .

1 Making algorithms precise.

2 Bits of recursion theory.

3 Church's Thesis.

4 The recursiveness of predicates, sets, properties, and relations.

5 ¢¦ And How It Is Represented in Typographical Number Theory.

1 Introspection and representation.

2 The representability of properties, relations, and functions ¢¦ .

3 ¢¦ and the G¶delian loop.

6 "I Am Not Provable".

1 Proof pairs.

2 The property of being a theorem of TNT (is not recursive!)

3 Arithmetizing substitution.

4 How can a TNT sentence refer to itself?

5 ³

6 Fixed point.

7 Consistency and omega-consistency.

8 Proving G1.

9 Rosser's proof.

7 The Unprovability of Consistency and the "Immediate Consequences" of G1 and G2.

1 G2.

2 Technical interlude.

3 "Immediate consequences" of G1 and G2.

4 Undecidable1 and undecidable2.

5 Essential incompleteness, or the syndicate of mathematicians.

6 Robinson Arithmetic.

7 How general are G¶del's results?

8 Bits of Turing machine.

9 G1 and G2 in general.

10 Unexpected fish in the formal net.

11 Supernatural numbers.

12 The culpability of the induction scheme.

13 Bits of truth (not too much of it, though).

Part II: The World after G¶del.

8 Bourgeois Mathematicians! The Postmodern Interpretations.

1 What is postmodernism?

2 From G¶del to Lenin.

3 Is "Biblical proof" decidable?

4 Speaking of the totality.

5 Bourgeois teachers!

6 (Un)interesting bifurcations.

9 A Footnote to Plato.

1 Explorers in the realm of numbers.

2 The essence of a life.

3 "The philosophical prejudices of our times".

4 From G¶del to Tarski.

5 Human, too human.

10 Mathematical Faith.

1 "I'm not crazy!"

2 Qualified doubts.

3 From Gentzen to the Dialectica interpretation.

4 Mathematicians are people of faith.

11 Mind versus Computer: G¶del and Artificial Intelligence.

1 Is mind (just) a program?

2 "Seeing the truth" and "going outside the system".

3 The basic mistake.

4 In the haze of the transfinite.

5 "Know thyself": Socrates and the inexhaustibility of mathematics.

12 G¶del versus Wittgenstein and the Paraconsistent Interpretation.

1 When geniuses meet ¢¦ .

2 The implausible Wittgenstein.

3 "There is no metamathematics".

4 Proof and prose.

5 The single argument.

6 But how can arithmetic be inconsistent?

7 The costs and benefits of making Wittgenstein plausible.

Epilogue.

References.

Index.