Mathematical Logic

  1. Set Theory
    1. Axiomatic Set Theory
      1. Foundations of Set Theory
        1. Naive Set Theory
          1. Limitations and Paradoxes
            1. Russell's Paradox
              1. Cantor's Paradox
            2. Zermelo-Fraenkel Set Theory (ZF)
              1. Axioms of ZF
                1. Axiom of Extensionality
                  1. Axiom of Regularity (Foundation)
                    1. Axiom of Pairing
                      1. Axiom of Union
                        1. Axiom of Power Set
                          1. Axiom of Infinity
                            1. Axiom of Replacement
                              1. Axiom of Empty Set
                                1. Axiom Schema of Separation
                                2. Role in Avoiding Paradoxes
                                3. Axiom of Choice (AC)
                                  1. Various Formulations and Equivalents
                                    1. Zorn's Lemma
                                      1. Well-Ordering Theorem
                                      2. Implications and Controversies
                                        1. Applications in Mathematics
                                        2. Continuum Hypothesis (CH)
                                          1. Statement and History
                                            1. Relationship with ZF and AC
                                              1. Independence from ZF
                                            2. Ordinals and Cardinals
                                              1. Ordinal Numbers
                                                1. Definition and Construction
                                                  1. Well-ordered Sets
                                                    1. Transfinite Induction and Recursion
                                                    2. Arithmetic of Ordinals
                                                      1. Ordinal Addition, Multiplication, Exponentiation
                                                      2. Limit Ordinals and Successor Ordinals
                                                      3. Cardinal Numbers
                                                        1. Cardinality of Sets
                                                          1. Definition and Properties
                                                            1. Cardinal Arithmetic
                                                              1. Cardinal Addition, Multiplication, Exponentiation
                                                              2. Infinite Cardinals
                                                                1. Aleph Numbers
                                                                  1. Comparison with Continuum
                                                              3. Independence Results
                                                                1. Gödel's Constructible Universe (L)
                                                                  1. Construction and Properties
                                                                    1. Role in Proving Consistency
                                                                    2. Cohen's Method of Forcing
                                                                      1. Basic Concepts and Techniques
                                                                        1. Applications in Proving Independence
                                                                          1. Examples: CH and AC
                                                                        2. Large Cardinals
                                                                          1. Concept and Motivation
                                                                            1. Hierarchies of Large Cardinals
                                                                              1. Measurable Cardinals
                                                                                1. Inaccessible Cardinals
                                                                                  1. Strong and Supercompact Cardinals
                                                                                  2. Impact on Set Theory and Mathematics
                                                                                    1. Consistency Strength
                                                                                      1. Role in Exploring Foundations of Mathematics
                                                                                  3. Model Theory
                                                                                  First Page
                                                                                  5. Computability Theory