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Characterizing All Models in Infinite Cardinalities [PhD Thesis]

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Author(s): Lauri Keskinen
Series: ILLC Dissertation Series DS-2011-05
Publisher: University of Amsterdam
Year: 2011

Language: English
Pages: 97
City: Amsterdam

1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 The logics Ln . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 In nitary second order languages . . . . . . . . . . . . . . 11
1.2.3 The constructible universe L . . . . . . . . . . . . . . . . . 14
1.2.4 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Ajtai's result, the countable case 19
2.1 A(L2; !) and L2-de nable well-order of the reals . . . . . . . . . . 19
2.2 Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Failure of A(L2; !) . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 The Frasse Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Submodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Fourth order logic 35
3.1 Coding subsets by collapsing cardinals . . . . . . . . . . . . . . . 35
3.2 Solovay's result on complete second order sentences . . . . . . . . 40
4 Generalized quanti ers 45
4.1 The countable case . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 The uncountable case . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 In nitary second order languages 51
5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2 Regular cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.4 Singular cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.5 A(L2;!; ) at a measurable cardinal . . . . . . . . . . . . . . . . . 61
6 A(L2; !) and large cardinal axioms 63
6.1 Large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2 Forcing axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7 Summary and future work 71
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Bibliography 75
Index 79
Samenvatting 81
Abstract 83