Sacks forcing, Laver forcing, and Martin's axiom
type: Item
instance of: scholarly article
title: Sacks forcing, Laver forcing, and Martin's axiom
zbMATH Open document ID: 0755.03026
DOI: 10.1007/BF01269943
author: Haim Judah
author: Arnold W. Miller
author: S. Shelah
published in: Archive for Mathematical Logic
publication date: 1992-09-26T00:00:00Z
review text: This paper is mainly about certain cardinals associated with Marczewski's ideal \(s^ 0\). The cardinals considered here include \(\text{add}(s^ 0)\) and \(\text{cov}(s^ 0)\), which denote the least cardinal \(\kappa\) such that the ideal \(s^ 0\) is not \(\kappa^ +\)-additive and the least \(\kappa\) such that \(2^ \omega\) can be covered by \(\kappa\) sets in \(s^ 0\), respectively. Some of the theorems proved about these cardinals are as follows: (a) \(\text{MA}+\neg\text{CH}\) implies that \(\text{add}(s^ 0)\) is the least cardinal \(\kappa\) such that \(p\Vdash_{\text{Sacks forcing}}\) ``\(\text{cof}(\mathfrak c)=\kappa\)'' for some \(p\); (b) If one adds \(\omega_ 2\) Sacks reals iteratively with countable supports to a model of CH, then \(\text{add}(s^ 0)=\omega_ 1\) and \({\mathfrak c}=\omega_ 2=\text{cov}(s^ 0)\) in the extension; (c) There is a model of \(\text{MA}+\neg\text{CH}\) in which \(\text{add}(s^ 0)=\text{cov}(s^ 0)=\omega_ 1\). Theorem (c) has also been obtained independently by \textit{B. Velickovic} [Compos. Math. 79, 279-294 (1991; Zbl 0735.03023)]. It follows from (a) and (c) that it is consistent with \(\text{MA}+\neg\text{CH}\) for Sacks forcing to collapse cardinals. By way of contrast, the authors include the theorem that MA implies that Laver forcing preserves all cardinals. There is an appendix in which it is shown that if \({\mathfrak c}=\omega_ 2\), then Sacks forcing adds a stationary subset of \(\omega_ 2\) that does not have a ground model stationary subset.
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E40
Mathematics Subject Classification ID: 03E50
zbMATH DE Number: 55776
zbMATH Keywords: additivity
zbMATH Keywords: Martin's axiom
zbMATH Keywords: cardinals associated with Marczewski's ideal \(s^ 0\)
zbMATH Keywords: Sacks forcing
zbMATH Keywords: Laver forcing
Wikidata QID: Q114693987
reviewed by: J. Takahashi
MaRDI profile type: Publication
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cites work: Q3792924
cites work: Q4175602
cites work: Iterated perfect-set forcing
cites work: Near coherence of filters. III: A simplified consistency proof
cites work: Singular sets and Baire order
cites work: Q3496430
cites work: The Ramsey sets and related sigma algebras and ideals
cites work: Classical theory of totally imperfect spaces
cites work: Variations on Lusin's Theorem
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description: scientific article; zbMATH DE number 55776