Recall that an inaccessible cardinal $\kappa$ is a Woodin cardinal if for every $A\subseteq V\_\kappa$ there is an unbounded set in $\kappa$ of $\lambda$ such that $V\_\kappa\models\lambda$ is $A$-strong.


This raises the question, of whether or not intermediate notions have been defined in the literature, and what sort of results can they be used to obtain?


For example, $\Sigma^n\_m$-Woodin is an inaccessible cardinal $\kappa$ such that for every $A$ which is $\Sigma^n\_m$-definable over $V\_\kappa$, there is an unbounded set of $\lambda<\kappa$ such that $V\_\kappa\models\lambda$ is $A$-strong.


It seems probable that there is such hierarchy of definable Woodin-ness, and that it might be used to obtain partial determinacy (and saturation) results.



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> Do these notions exist in the literature, and if so do they have interesting consequences?
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"Weakly" Woodin cardinals
(As I pointed out in a comment) yes, partial Woodinness is common in arguments in inner model theory. Accordingly, you obtain determinacy results addressing specific pointclasses (typically, well beyond projective). To illustrate this, let me "randomly" highlight two examples:


* See [here](https://andrescaicedo.wordpress.com/2009/01/19/set-theory-seminar-quasiiterations/) for $\Sigma^1\_2$-Woodin cardinals and, more generally, the notion of a cardinal $\delta$ being Woodin with respect to a family $\frak A\subseteq \mathcal P(\delta)$.
* See [here](https://math.berkeley.edu/~steel/papers/mousejul04.pdf) for $\Gamma$-Woodin cardinals (and coarse mice) for $\Gamma$ a good pointclass.


There are also more recent examples, of course.


