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June 2008 Cardinal invariants and the collapse of the continuum by Sacks forcing
Miroslav Repický
J. Symbolic Logic 73(2): 711-727 (June 2008). DOI: 10.2178/jsl/1208359068

Abstract

We study cardinal invariants of systems of meager hereditary families of subsets of ω connected with the collapse of the continuum by Sacks forcing 𝕊 and we obtain a cardinal invariant 𝔥ω such that 𝕊 collapses the continuum to 𝔥ω and 𝔥≤𝔥ω≤𝔟. Applying the Baumgartner-Dordal theorem on preservation of eventually narrow sequences we obtain the consistency of 𝔥=𝔥ω<𝔟. We define two relations ≼*0 and ≼*1 on the set (ωω)Fin of finite-to-one functions which are Tukey equivalent to the eventual dominance relation of functions such that if ℱ⊆(ωω)Fin is ≼*1-unbounded, well-ordered by ≼*1, and not ≼*0-dominating, then there is a nonmeager p-ideal. The existence of such a system ℱ follows from Martin’s axiom. This is an analogue of the results of [3], [9,10] for increasing functions.

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Miroslav Repický. "Cardinal invariants and the collapse of the continuum by Sacks forcing." J. Symbolic Logic 73 (2) 711 - 727, June 2008. https://doi.org/10.2178/jsl/1208359068

Information

Published: June 2008
First available in Project Euclid: 16 April 2008

zbMATH: 1156.03048
MathSciNet: MR2414473
Digital Object Identifier: 10.2178/jsl/1208359068

Subjects:
Primary: 03E17
Secondary: 03E35, 03E40, 03E50

Rights: Copyright © 2008 Association for Symbolic Logic

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Vol.73 • No. 2 • June 2008
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