I will sketch a proof that, assuming \(0^\dagger\) does not exist, if there is a partition of the real line \(\mathbb{R}\) into \(\aleph_\omega\) Borel sets, then there is also a partition of \(\mathbb{R}\) into \(\aleph_{\omega+1}\) Borel sets. (And the same is true for any singular cardinal of countable cofinality in place of \(\aleph_\omega\).) This contrasts starkly with the situation for successor-of-successor cardinals, where the spectrum of possible sizes of partitions of \(\mathbb{R}\) into Borel sets can seemingly be made completely arbitrary. For example, given any \(A \subseteq \omega\) with \(0, 1 \in A\), there is a forcing extension in which \(A = \{n < \omega :\) there is a partition of \(\mathbb{R}\) into \(\aleph_n\) Borel sets\(\}\).
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