Generic absoluteness and the continuum
type: Item
instance of: scholarly article
title: Generic absoluteness and the continuum
zbMATH Open document ID: 1028.03040
published in: Mathematical Research Letters
publication date: 2003-05-15T00:00:00Z
review text: Generic absoluteness for a class of forcings states that \((H(\aleph_2),\in)\prec_1 (V^B,\in)\) whenever \(V^B\) is a Boolean-valued extension of the universe by a complete Boolean algebra from that class. The main result of the paper is that generic absoluteness for Boolean-valued extensions that preserve stationary subsets of \(\omega_1\) implies the continuum has a well-order of length \(\omega_2\) (definable in \(H(\aleph_2)\) from a single \(\omega_1\)-sequence of reals). Given the one-to-one sequence \(\langle r_\alpha:\alpha\in\omega_1\rangle\) of elements of the Cantor set the coding of subsets of \(\omega\) by ordinals in \(\omega_2\) proceeds as follows. To every countable set of ordinals \(X\) one associates the real \(r_X=r_{\text{otp} X}\). Furthermore the first-difference function \(\Delta(x,y)=\min\{n:x(n)\neq y(n)\}\) on the Cantor set is employed to associate (potentially) to every subset \(S\) of \(\omega_1\) three ordinals \(\gamma>\beta>\alpha\geq\omega_1\) where \(\gamma\) is written as the increasing union of a continuous and increasing sequence \(\langle N_\nu:\nu\in\omega_1\rangle\) of countable sets, all such that \(N_\nu\cap\omega_1\in S\) if and only if \(\Delta(r_{N_\nu\cap\alpha},r_{N_\nu\cap\beta})\) is the largest of the values \(\Delta(u,v)\), where \(u\) and \(v\) can be \(r_{N_\nu\cap\alpha}\), \(r_{N_\nu\cap\beta}\) and \(r_{N_\nu}\), respectively. Now if \((\gamma,\beta,\alpha)\) were associated to two sets \(S\) and \(T\) in this way, with sequences \(\langle N_\nu\rangle_\nu\) and \(\langle M_\nu\rangle_\nu\) respectively, then, since \(N_\nu=M_\nu\) for a cub set of \(\nu\), one would find that \(S\) and \(T\) were equal modulo the cub filter. As one can find \(2^{\aleph_1}\) inequivalent sets, the possibility of assigning a triple and sequence to every subset of \(\omega_1\), abbreviated \(\theta_{\text{AC}}\) in the paper, would imply \(2^{\aleph_1}=\aleph_2\). It turns that one can find for each \(S\) a generic extension in …
reviewed by: K. P. Hart
Mathematics Subject Classification ID: 03E50
Mathematics Subject Classification ID: 03E40
Mathematics Subject Classification ID: 03E65
zbMATH DE Number: 1886041
zbMATH Keywords: generic absoluteness
zbMATH Keywords: \(H(\aleph_2)\)
zbMATH Keywords: Continuum Hypothesis
zbMATH Keywords: Boolean-valued extensions
author: Stevo Todorčević
MaRDI profile type: Publication
DOI: 10.4310/MRL.2002.V9.N4.A6
description: scientific article; zbMATH DE number 1886041