The Covering Property Axiom, CPA: A Combinatorial Core of the Iterated Perfect Set Model
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 self contained and there are natural applications to real analysis and topology. Researchers who use set theory in their work will find much of interest in this book.
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Prisms and axioms CPA Saint and CPA prism
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addition apply arbitrary argument assume assumption axiom belongs Borel cardinality chapter choose Claim Clearly clopen closed comeager compact condition consider consistent construct contains continuous function contradicting Corollary countable covering CPA game CPA prism CPAcube defined definition dense density desired easy elements enumeration equal example exists extending F-independent fact finish finite forcing function f holds homeomorphic ideal implies independent induction infinite intersects iterated perfect set Lemma less maximal meager measure zero Moreover nonempty Note notice notion pairwise disjoint particular Perf(X perfect cube Player points Polish space presented PROOF PROOF OF THEOREM Proposition prove relation Remark result satisfies selective sequence strategy subcube subprism subset Theorem ultrafilter uniformly union winning