Mathematical Logic

  1. Incompleteness Theorems
    1. Formal Theories of Arithmetic
      1. Robinson Arithmetic
        1. Axioms and Properties
          1. Undecidability
          2. Peano Arithmetic
            1. First-Order Axioms
              1. Induction Schema
                1. Strength and Limitations
                2. Non-Standard Models of Arithmetic
                  1. Existence and Properties
                    1. Overspill Principle
                  2. Representability of Computable Functions
                    1. Definition of Representability
                      1. Examples in Arithmetic
                        1. Representability Theorem
                        2. Gödel Numbering
                          1. Encoding Formulas and Proofs
                            1. Properties of Gödel Numbering
                              1. Primitive Recursive Relations
                              2. The Diagonal Lemma
                                1. Statement and Proof Outline
                                  1. Role in Incompleteness Proofs
                                    1. Fixed Point Theorem
                                    2. Gödel's First Incompleteness Theorem
                                      1. Construction of the Gödel Sentence
                                        1. Proof of the Theorem
                                          1. Consequences for Formal Systems
                                            1. Incompleteness of Arithmetic
                                              1. Limitations of Formalization
                                              2. Rosser's Improvement
                                              3. Gödel's Second Incompleteness Theorem
                                                1. Formalizing Consistency
                                                  1. The Unprovability of Consistency
                                                    1. Hilbert's Program
                                                      1. Derivability Conditions

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                                                    7. Axiomatic Set Theory