Proves consistency of boldface Σ¹₁-separation on ω₁^ω₁ via forcing from L that preserves CH.
A Failure of $\Pi^1_{n+3}$-Reduction in the Presence of $\Sigma^1_{n+3}$-Separation
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abstract
We show that one can force over $L$ that $\Sigma^1_3$-separation holds, while $\Pi^1_3$-reduction fails, thus separating these two principles for the first time. The construction can be lifted to canonical inner models $M_n$ with $n$-many Woodin cardinals, yielding that assuming the existence of $M_n$, $\Sigma^1_{n+3}$-separation can hold, yet $\Pi^1_{n+3}$-reduction fails.
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math.LO 1years
2026 1verdicts
UNVERDICTED 1representative citing papers
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Forcing $\mathbf{\Sigma}^1_1$-Separation on $\omega_1^{\omega_1}$
Proves consistency of boldface Σ¹₁-separation on ω₁^ω₁ via forcing from L that preserves CH.