In [the proof that Martin's maximum implies (\*)](https://ivv5hpp.uni-muenster.de/u/rds/MM_implies_star_final_version_March_18_2021.pdf), the introduction gives the following theorem:



> 
> Assume there is a proper class of Woodin cardinals. Then, the following are equivalent:
> 
> 
> * $(^\*)$
> * For any $\Pi\_2$ $\mathcal{L}$-sentence $\sigma$, where $\mathcal{L}$ is the language of the structure $(H\_{\omega\_2}; \in, \mathrm{NS}\_{\omega\_1}, \mathcal{A})\_{\mathcal{A} \in \mathcal{P}(\mathbb{R}) \cap L(\mathbb{R})}$, if $\sigma$ is $\Omega$-consistent, then $\sigma$ is true.
> 
> 
> 


Now, let's define the axiom $(^\*)^+$ to be the conjunction of the following two statements:


* There exists a proper class of Woodin cardinals.
* For any $\mathcal{L}$-sentence $\sigma$, where $\mathcal{L}$ is the language of the structure $(H\_{\omega\_2}; \in, \mathrm{NS}\_{\omega\_1}, \mathcal{A})\_{\mathcal{A} \in \mathcal{P}(\mathbb{R}) \cap L(\mathbb{R})}$, if $\sigma$ is $\Omega$-consistent, then $\sigma$ is true.


Namely, the restriction to $\Pi\_2$-sentences (statements of the form $\forall x \exists y \varphi(x,y)$) is removed. Now, here's my question.



> 
> Is $(^\*)^+$ inconsistent? If it is consistent (assuming the consistency of certain large cardinal assumptions), then is it equiconsistent with the regular axiom $(^\*)^+$, or strictly stronger?
> 
> 
> 



An extension of Woodin's star axiom
This principle is inconsistent, even if we just look at $(H\_{\omega\_2};\in)$. This is because - for example - the continuum hypothesis is expressible as a sentence in this structure ("There is an $\omega\_1$-sequence of reals such that every real appears in the sequence"), but both $\mathsf{CH}$ and $\neg\mathsf{CH}$ are $\Omega$-consistent.


Indeed, the example of $\mathsf{CH}$ shows that $\Pi\_2$ is basically the best we can hope for: already at $\Pi\_2\vee\Sigma\_2$ we get inconsistency.


