Lecture | Topics |
---|---|

Lecture #1 | Partial orders. The Rasiowa-Sikorski Lemma. Every countable dense linearly ordered set without endpoints is order-isomorphic to the rational line. |

Lecture #2 | Transitive sets. Well-orders are rigid. The class of ordinals is well-ordered. |

Lecture #3 | Every two well-ordered sets are compatible. Finite ordinals. Every well-ordered set is isomorphic to a unique ordinal. Ordinals arithmetic: sums and multiplications. |

Lecture #4 | Transfinite induction, and applications to ordinal arithmetic. Ordinal power. |

Lecture #5 | Hartogs' number. Definition by recursion. The axiom of choice. Zermelo's well-ordering principle. Hausdorff's maximality principle. Zorn's lemma. Hartogs' comparability of cardinals. Tychonoff's theorem. |

Lecture #6 | The axioms of ZFC. The effect of AC on infinite graph theory: the Shelah-Soifer graph, and the Komjath-Galvin graph. |

Lecture #7 | ZF implies that Hartogs' number is indeed a set. Cardinals, and cardinal arithmetic. Cantor's theorem. |

Lecture #8 | kxk=k for every infinite cardinal k, and applications to cardinal arithmetic. Successor and limit cardinals. The Aleph function. Cofinality of partially ordered-sets. Regular and singular cardinals. Every successor cardinal is regular. |

Lecture #9 | Analysis of the Aleph function: cofinality, surjectivity, fixed points. Some more cardinal arithmetic, including Konig's lemma and Hausdorff's formula. |

Lecture #10 | Cardinal arithmetic assuming GCH. Strong limit cardinals and the Beth function. Ideals. The nonstationary ideal is sigma-additive. Normal functions. Closed and unbounded sets. Appendix on Ramsey's theorem. |

Lecture #11 | Filters. CUB is the dual of NS. Diagonal union. The nonstationary ideal is normal. Fodor's lemma. Ulam's matrix. |

Lecture #12 | Every stationary subset of w1 may be partitioned into w1 many pairwise disjoint sets. Partition calculus for dimension one. Partition calculus for dimension two: the Sierpinski coloring. The Delta-system lemma. |

Lecture #13 | Set-theoretic implementation of the classic system of numbers: natural numbers, integers, rational numbers, the real line (Dedekind cuts). Order-theoretic uniqueness of the real line. Cardinality of the real line. |

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