I want to proof that the Borel hierarchy on $\mathbb{R}$ collapses after $\omega_1$ steps. However, I'm stuck and don't know where to begin, I suspect I need to use the fact that $\aleph_1$ is a regular cardinal and the first uncountable cardinal. I'd like to know some hints on where to continue. Thanks.
2026-03-30 10:38:39.1774867119
Proof that the Borel hierarchy on $\mathbb{R}$ collapses after $\omega_1$ steps.
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