CONTENTS
Prefaceto the second edition............................................... ix
Prefaceto the first edition ................................................. xi
About this book ............................................................ xiii
Introduction ............................................................... 1
Chapter 1. The basic classicalnotions.................................... 9
1A. Perfect Polish spaces .............................................. 9
1B. The Borel pointclasses of finite order............................... 13
1C. Computing with relations; closure properties ....................... 18
1D. Parametrization and hierarchy theorems............................ 26
1E. The projective sets ................................................ 29
1F. Countable operations ... ......................................... 33
1G. Borel functions and isomorphisms ................................. 37
1H. Historical and other remarks ...................................... 46
Chapter 2. κ-Suslin and -Borel .......................................... 49
2A. The Cantor-Bendixson Theorem................................... 50
2B. κ-Suslin sets...................................................... 51
2C. Trees and the Perfect Set Theorem ................................. 57
2D. Wellfounded trees................................................. 62
2E. The Suslin Theorem............................................... 65
2F. Inductive analysis of projections of trees............................ 70
2G. The Kunen-Martin Theorem ...................................... 74
2H. Category and measure............................................. 79
2I. Historical remarks ................................................ 85
Chapter 3. Basic notions of the effectivetheory.......................... 87
3A. Recursive functions on the integers................................. 89
3B. Recursive presentations............................................ 96
3C. Semirecursive pointsets............................................ 101
3D. Recursive and Γ-recursive functions................................ 110
3E. The Kleene pointclasses........................................... 118
3F. Universal sets for the Kleene pointclasses...........................125
3G. Partial functions and the substitution property ..................... 130
3H. Codings, uniformity and good parametrizations.................... 135
3I. Effective theory on arbitrary (perfect) Polish spaces.................141
vii
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