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In the article Projectively wellordered inner models, Steel proves the following theorem (4.12):

Theorem: Let $n < \omega$ and suppose $\mathcal{M}_n^{\sharp}$ exists. Let $\delta=\delta^1_{n+2}$. Then for any $A \subseteq \omega$, $A$ is $\Pi^1_{n+2}$ if and only if $A$ is $\Sigma_1$ over $\mathcal{J}^{\mathcal{M}}_{\delta}$ if $n$ is odd and $A$ is $\Sigma^1_{n+2}$ if and only if $A$ is $\Sigma_1$ over $\mathcal{J}^{\mathcal{M}}_{\delta}$ if $n$ is even.

In the theorem it is implicit that $\Sigma_1$ means $\Sigma_1^{HC}$. The result is an extension of the so-called "Spector-Gandy companion theorem".

I was wondering if the result can be extended to sets $A \subseteq \omega^{\omega}$: for example can one have $A \subseteq \omega^{\omega}$ is $\Pi^1_{n+2}$ if and only if $A$ is $\Sigma_1$ over $\mathcal{J}^{\mathcal{M}}_{\delta}$, where $\Sigma_1$ would be $\Sigma_1^A$, for some appropriate set $A$ ? (this would be the case where $n$ is odd). Notice this is in the lightface setting so the formulas would not involve any real numbers as parameters.

Secondly, can the result be extended to the boldface context: $A \subseteq \omega^{\omega}$ is $\bf\Pi^1_{n+2}$ if and only if $A$ is $\Sigma_1$ over $\mathcal{J}^{\mathcal{M}}_{\delta}$, where $\Sigma_1$ would be $\Sigma_1^A$, for some appropriate set $A$ (case n is odd)? In this case formulas would involve real numbers as parameters.

Let me recall some definitions: $\mathcal{M}_n^{\sharp}$ is the mouse which has $n$ Woodin cardinals and a top measure (the sharp). It has a unique $(\omega, \omega_1)$ iteration strategy. $\delta^1_{n+2}$ is the supremum of the length of the $\Delta^1_{n+2}$ prewellorderings.

Edit: Following Andres suggestion, the $\mathcal{M}$ above should be taken to be $\mathcal{M}_n$, the mouse with $n$ Woodin cardinals.

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  • $\begingroup$ What is $\mathcal{J}_\delta^\mathcal{M}$? $\endgroup$
    – Noah Schweber
    Commented Mar 27, 2014 at 3:04
  • $\begingroup$ $\mathcal{M}$ is a mouse so it has an extender sequence and it is built using Jensens' $J$ hierarchy (this is what the $\mathcal{J}$ refers to). $\mathcal{J}^{\mathcal{M}}_{\delta}$ is an initial segment of $\mathcal{M}$ up to $\delta$ for $\delta < o(\mathcal{M})$. So basically, putting $\mathcal{M}=\mathcal{J}^{\bf E}_{\alpha}$, where $\bf E$ is the extender sequence of $\mathcal{M}$ and $\alpha=o(\mathcal{M})$ then $\mathcal{J}^{\mathcal{M}}_{\delta}=\mathcal{J}^{\bf E}_{\delta}$. $\endgroup$
    – Rachid Atmai
    Commented Mar 27, 2014 at 3:34
  • $\begingroup$ So $\mathcal{J}^{\bf E}_{\alpha}$ denotes the structure $(J^{\bf E}_{\alpha}, \bf {E}|\alpha$, $E_{\alpha})$, $\bf {E}|\alpha$ is the extender sequence chopped at $\alpha$ and $E_{\alpha}$ is the $\alpha$-th extender on the sequence. I probably should add this to the above for clarity. $\endgroup$
    – Rachid Atmai
    Commented Mar 27, 2014 at 3:39
  • $\begingroup$ You are misquoting the paper. It is not an arbitrary $\mathcal M$ used here, but rather $\mathcal M_n$, the result of iterating the sharp extender of $\mathcal M_n^\sharp$ through the ordinals. $\endgroup$
    – Andrés E. Caicedo
    Commented Mar 27, 2014 at 21:03
  • $\begingroup$ The intended generalization is false: $\mathcal M_n$ only has countably many reals. One can work with codes (definitions) for projective sets rather than the sets themselves to avoid this, and work not with $\mathcal M_n^\sharp$ but the relativized versions $\mathcal M_n^\sharp(x)$. The appropriate correctness results in this context are also addressed in section 4 of Steel's paper or, if you prefer a somewhat more modern presentation, see section 7 of his Handbook article. $\endgroup$
    – Andrés E. Caicedo
    Commented Mar 27, 2014 at 21:15

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