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According to Lightface mice with finitely many Woodin cardinals from optimal determinacy hypotheses by Yizheng Zhu, theorem 1.1, over $\mathbf{Σ^1_{2n+1}}$ determinacy, $Δ^1_{2n+2}$ determinacy is equivalent to existence of an inner model with $2n+1$ countable Woodin cardinals (which is a $Σ^1_3$ statement). (Per the theorem, it is also equivalent to existence of (countably iterable) $M_{2n+1}^\#$.)

However, $Δ^1_{2n+2}$ determinacy is a $Σ^1_{2n+3}$ statement, and it seems suspicious that over $\mathbf{Σ^1_{2n+1}}$ determinacy, it would be equivalent to a $Σ^1_3$ statement.

Am I understanding this part of this theorem correctly?
Have there been any technical rebuttals to the claim?
If yes, what is the correct statement?
If no, what is the intuition behind it being equivalent to a $Σ^1_3$ statement?

Improve this question
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  • $\begingroup$ If a non-mathematician reads the first sentence, he/she certainly doesn't imagine at all the degree of virtuality of these degenerate mice, even in Rome... $\endgroup$
    – Jean Marie Becker
    Commented Jan 30, 2019 at 18:57
  • $\begingroup$ In inner model theory, mice are analogs of levels of the constructible hierarchy. Like ordinary mice, the mice of interest in set theory tend to be small, and there are many of them. $\endgroup$
    – Dmytro Taranovsky
    Commented Sep 22, 2019 at 0:59

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