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The relative consistency of some consequences of the existence of measurable cardinal numbers

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Bowen K. A.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS Introduction..................................................................................................................................................5 I. Topological forcing..................................................................................................................................8 1.1. Preliminaries.......................................................................................................................................8 1.2. Languages, satisfaction, and relative constructibility..................................................................10 1.3. Forcing..................................................................................................................................................15 1.4. Adding a set of generic sets.............................................................................................................26 1.5. Adding a proper class of generic sets...........................................................................................26 1.6. Collapsing cardinals.........................................................................................................................33 II. Reflection principles and their consequences................................................................................39 2.1. Reflection principles..........................................................................................................................39 2.2. The Tree and Ramsey properties...................................................................................................46 III. Application to relative consistency results.......................................................................................50 3.1. The main result...................................................................................................................................50 3.2. A converse consistency result..........................................................................................................54 References..................................................................................................................................................58

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1 Bowen K. A.

1 1972

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The relative consistency of some consequences of the existence of measurable cardinal numbers