The covering property axiom, CPA: A combinatorial core of the iterated perfect set model

Krzysztof Ciesielski, Janusz Pawlikowski0521839203, 9780521839204, 9780511217401

Here the authors formulate and explore a new axiom of set theory, CPA, the Covering Property Axiom. CPA is consistent with the usual ZFC axioms, indeed it is true in the iterated Sacks model and actually captures the combinatorial core of this model. A plethora of results known to be true in the Sacks model easily follow from CPA. Replacing iterated forcing arguments with deductions from CPA simplifies proofs, provides deeper insight, and leads to new results. One may say that CPA is similar in nature to Martin’s axiom, as both capture the essence of the models of ZFC in which they hold. The exposition is a self contained and there are natural applications to real analysis and topology. Researchers that use set theory in their work will find much of interest in this book.

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Table of contents :
Cover……Page 1
Half-title……Page 3
Title……Page 7
Copyright……Page 8
Dedication……Page 9
Contents……Page 11
Overview……Page 13
Preliminaries……Page 21
1 Axiom CPA and its consequences: properties (A)–(E)……Page 25
1.1 Perfectly meager sets, universally null sets, and continuous images of sets of cardinality continuum……Page 28
1.2 Uniformly completely Ramsey null sets……Page 33
1.3 cof(N) = Omega1……Page 35
1.4 Total failure of Martin’s axiom……Page 36
1.5 Selective ultrafilters and the reaping numbers r and rSigma……Page 39
1.6 On the convergence of subsequences of real-valued functions……Page 41
1.7 Some consequences of cof(N) = Omega1: Blumberg’s theorem, strong measure zero sets, magic sets, and the cofinality of Boolean algebras……Page 45
1.8 Remarks on a form and consistency of the axiom CPAcube……Page 51
2 Games and axiom CPA……Page 55
2.1 CPA and disjoint coverings……Page 56
2.2 MAD families and the numbers a and r……Page 58
2.3 Uncoun table γ-sets and strongly meager sets……Page 60
2.4 Nowhere meager set A × A…R2 intersecting continuous functions on a small set……Page 69
2.5 Remark on a form of CPA……Page 72
3 Prisms and axioms CPA and CPA……Page 73
3.1 Fusion for prisms……Page 78
3.2 On F-independent prisms……Page 82
3.3 CPA, additivity of s0, and more on (A)……Page 93
3.4 Intersections of Omega1 many open sets……Page 97
3.5 α-prisms and separately nowhere constant functions……Page 102
3.6 Multi-games and other remarks on CPA and CPA……Page 112
4 CPA and coverings with smooth functions……Page 115
4.1 Chapter overview; properties (H) and (R)……Page 116
4.2 Proof of Proposition 4.1.3……Page 121
4.3 Proposition 4.2.1: a generalization of a theorem of Morayne……Page 124
4.4 Theorem 4.1.6: on………Page 127
4.5 Examples related to the cov operator……Page 129
5.1 Nice Hamel bases……Page 134
5.2 Some additive functions and more on Hamel bases……Page 140
5.3 Selective ultrafilters and the number u……Page 152
5.4 Nonselective P-points and number i……Page 157
5.5 Crowded ultrafilters on Q……Page 163
6 CPA and properties (F) and (G)……Page 167
6.1 …and many ultrafilters……Page 169
6.2 Surjections onto nice sets must be continuous on big sets……Page 171
6.3 Sums of Darboux and continuous functions……Page 172
6.4 Remark on a form of CPA……Page 178
7.1 Notations and basic forcing facts……Page 179
7.2 Consistency of CPA……Page 183
Notation……Page 186
References……Page 189
Index……Page 196