# Measuring club-sequences together with the continuum large

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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.

### Details

Original language English 1066-1079 14 Journal of Symbolic Logic 82 3 8 Sep 2017 https://doi.org/10.1017/jsl.2017.4 Published - Sep 2017 Yes

### Research areas

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

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ID: 101664572

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