Measuring club-sequences together with the continuum large

David Asperó, Miguel Angel Mota

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Abstract

Measuring says that for every sequence $(C_\delta)_{\delta<\omega_1}$ with each $C_\delta$ being a closed subset of $\delta$ there is a club $C\subseteq\omega_1$ such that for every $\delta\in C$,a tail of $C\cap\delta$ is either contained in or disjoint from $C_\delta$. We answer a question of Justin Moore by building a forcing extension satisfying measuring together with $2^{\aleph_0}>\aleph_2$. The construction works over any model of ZFC + CH and can be described as a finite support forcing iteration with systems of countable models as side conditions and with symmetry constraints imposed on its initial segments. One interesting feature of this iteration is that it adds dominating functions $f:\omega_1\longrightarrow\omega_1$ mod. countable at each of its stages.
Original languageEnglish
Pages (from-to)1066-1079
Number of pages14
JournalJournal of Symbolic Logic
Volume82
Issue number3
Early online date8 Sep 2017
DOIs
Publication statusPublished - Sep 2017

Keywords

  • Measuring
  • large continuum
  • iterated forcing with symmetric systems of models as side conditions

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