## Combinatorial Set Theory: With a Gentle Introduction to ForcingThis book provides a self-contained introduction to modern set theory and also opens up some more advanced areas of current research in this field. The first part offers an overview of classical set theory wherein the focus lies on the axiom of choice and Ramsey theory. In the second part, the sophisticated technique of forcing, originally developed by Paul Cohen, is explained in great detail. With this technique, one can show that certain statements, like the continuum hypothesis, are neither provable nor disprovable from the axioms of set theory. In the last part, some topics of classical set theory are revisited and further developed in the light of forcing. The notes at the end of each chapter put the results in a historical context, and the numerous related results and the extensive list of references lead the reader to the frontier of research. This book will appeal to all mathematicians interested in the foundations of mathematics, but will be of particular use to graduates in this field. |

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### Contents

1 | |

8 | |

From Martins Axiom to Cohens Forcing | 256 |

Combinatorics of Forcing Extensions | 355 |

438 | |

Names Index | 443 |

447 | |

### Common terms and phrases

arbitrary assume Axiom of Choice Baire property belongs bijection cardinal number Chapter Cohen forcing Cohen reals colouring combinatorial consistent with ZFC construct countable sets define denoted dense set disjoint dom(p element equivalent example exists fact filter fin(a finite set Firstly Fn(a forcing notion formula function f Fundam Furthermore Gödel ground model Halbeisen Hence implies induction infinite set infinite subset Jech LEMMA let G limit ordinal Logic mad family MAIDEN Math Mathias forcing meagre meagre sets model of ZFC n e a n e o natural numbers Notice open dense subset open set P-generic P-name P-points partially ordered set partition permutation model positive integers Prime Ideal proof PROPOSITION prove Ramsey ultrafilters RAMSEY’S THEOREM RELATED RESULT SAHARON SHELAH satisfies ccc seq(a sequence Set Theory Shelah stipulating transfinite uncountable well-ordered ωω