## Problems and Theorems in Classical Set TheoryAlthough the ?rst decades of the 20th century saw some strong debates on set theory and the foundation of mathematics, afterwards set theory has turned into a solid branch of mathematics, indeed, so solid, that it serves as the foundation of the whole building of mathematics. Later generations, honest to Hilbert’s dictum, “No one can chase us out of the paradise that Cantor has created for us” proved countless deep and interesting theorems and also applied the methods of set theory to various problems in algebra, topology, in?nitary combinatorics, and real analysis. The invention of forcing produced a powerful, technically sophisticated tool for solving unsolvable problems. Still, most results of the pre-Cohen era can be digested with just the knowledge of a commonsense introduction to the topic. And it is a worthy e?ort, here we refer not just to usefulness, but, ?rst and foremost, to mathematical beauty. In this volume we o?er a collection of various problems in set theory. Most of classical set theory is covered, classical in the sense that independence methods are not used, but classical also in the sense that most results come fromtheperiod,say,1920–1970.Manyproblemsarealsorelatedtoother?elds of mathematics such as algebra, combinatorics, topology, and real analysis. We do not concentrate on the axiomatic framework, although some - pects, such as the axiom of foundation or the role ˆ of the axiom of choice, are elaborated. |

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### Contents

III | 3 |

IV | 9 |

V | 13 |

VI | 15 |

VII | 19 |

VIII | 23 |

IX | 32 |

X | 37 |

XXXVII | 158 |

XXXVIII | 163 |

XXXIX | 173 |

XL | 185 |

XLI | 213 |

XLII | 223 |

XLIII | 236 |

XLIV | 265 |

XI | 43 |

XII | 51 |

XIII | 55 |

XIV | 58 |

XV | 63 |

XVI | 65 |

XVII | 67 |

XVIII | 70 |

XIX | 70 |

XX | 79 |

XXI | 81 |

XXII | 84 |

XXIII | 89 |

XXIV | 93 |

XXV | 95 |

XXVI | 100 |

XXVII | 107 |

XXVIII | 109 |

XXIX | 111 |

XXX | 116 |

XXXI | 123 |

XXXII | 126 |

XXXIII | 129 |

XXXIV | 133 |

XXXV | 135 |

XXXVI | 147 |

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### Common terms and phrases

0–1 sequences antichain arbitrary Assume axiom of choice Cardinal and Ordinal cardinality continuum cardinality smaller chromatic number claim clearly closed club set cofinal color contradiction countable set decomposition define densely ordered diﬀerent enumeration ﬁnite ﬁrst graph Hamel basis hence implies increasing sequence induction hypothesis inﬁnite infinite set intersection interval largest element lemma limit ordinal Math N-set natural numbers nonempty nonstationary open sets order type Ordinal Numbers pairwise disjoint partially ordered set points Polish Sci power continuum preceding problem proof prove Publ rational numbers real numbers regular cardinal set of cardinality Sierpihski similar smallest element solution stationary set subgraph subset successor ordinal suppose Suslin tree theorem topology transfinite recursion tree ultrafilter unbounded set uncountable vertex vertices Warszawa well-ordered set