**Introduction** |
1 |

**Itinerary I: Metatheoretical Properties of Axiomatic Systems** |
3 |

1.1 Introduction |
3 |

1.2 Peano’s school on the logical structure of theories |
4 |

1.3 Hilbert on axiomatization |
8 |

1.4 Completeness and categoricity in the work of Veblen and Huntington |
10 |

1.5 Truth in a structure |
12 |

**Itinerary II: Bertrand Russell’s Mathematical Logic** |
15 |

2.1 From the Paris congress to the *Principles of Mathematics* 1900–1903 |
15 |

2.2 Russell and Poincare on predicativity |
19 |

2.3 On Denoting |
21 |

2.4 Russell’s ramified type theory |
22 |

2.5 The logic of *Principia* |
25 |

2.6 Further developments |
26 |

**Itinerary III: Zermelo’s Axiomatization of Set Theory and Related Foundational Issues** |
29 |

3.1 The debate on the axiom of choice |
29 |

3.2 Zermelo’s axiomatization of set theory |
32 |

3.3 The discussion on the notion of “definit” |
35 |

3.4 Metatheoretical studies of Zermelo’s axiomatization |
38 |

**Itinerary IV: The Theory of Relatives and Löwenheim’s Theorem** |
41 |

4.1 Theory of relatives and model theory |
41 |

4.2 The logic of relatives |
44 |

4.3 Löwenheim’s theorem |
46 |

4.4 Skolem’s first versions of Löwenheim’s theorem |
56 |

**Itinerary V: Logic in the Hilbert School** |
59 |

5.1 Early lectures on logic |
59 |

5.2 The completeness of propositional logic |
60 |

5.3 Consistency and completeness |
61 |

5.4 Axioms and inference rules |
66 |

5.5 *Grundzüge der theoretischen Logik* |
70 |

5.6 The decision problem |
71 |

5.6.1 The decision problem in the tradition of algebra of logic |
72 |

5.6.2 Work on the decision problem after 1920 |
73 |

5.7 Combinatory logic and ?-calculus |
74 |

5.8 Structural inference: Hertz and Gentzen |
76 |

**Itinerary VI: Proof Theory and Arithmetic** |
81 |

6.1 Hilbert’s Program for consistency proofs |
81 |

6.2 Consistency proofs for weak fragments of arithmetic |
82 |

6.3 Ackermann and von Neumann on epsilon substitution |
87 |

6.4 Herbrand’s Theorem |
92 |

6.5 Kurt Gödel and the incompleteness theorems |
94 |

**Itinerary VII: Intuitionism and Many-valued Logics** |
99 |

7.1 Intuitionistic logic |
99 |

7.1.1 Brouwer’s philosophy of mathematics |
99 |

7.1.2 Brouwer on the excluded middle |
101 |

7.1.3 The logic of negation |
102 |

7.1.4 Kolmogorov |
103 |

7.1.5 The debate on intuitionist logic |
106 |

7.1.6 The formalization and interpretation of intuitionistic logic |
108 |

7.1.7 Gödel’s contributions to the metatheory of intuitionistic logic |
110 |

7.2 Many-valued logics |
111 |

**Itinerary VIII: Semantics and Model-theoretic Notions** |
117 |

8.1 Background |
117 |

8.1.1 The algebra of logic tradition |
117 |

8.1.2 Terminological variations (systems of objects, models, and structures) |
118 |

8.1.3 Interpretations for propositional logic |
119 |

8.2 Consistency and independence for propositional logic |
120 |

8.3 Post’s contributions to the metatheory of the propositional calculus |
123 |

8.4 Semantical completeness of first-order logic |
124 |

8.5 Models of first order logic |
129 |

8.6 Completeness and categoricity |
130 |

8.7 Tarski’s definition of truth |
134 |

Notes |
141 |

Bibliography |
149 |

Index of Citations |
175 |