# Partitioning the real line into Borel sets

03.11.2022 16:45 - 18:15

W. Brian (U of North Carolina at Charlotte, US)

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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