/sci/ - Science & Math

>>11530615

A theorem of Martin and Harrington linking determinancy to large cardinals.
[math]\Sigma_1^1(x)[/math]-determinancy if and only if [math]x^{\sharp}[/math] exists for every [math]x\subseteq\omega[/math]. In particular, if there exists a measurable cardinal then [math]\mathbf{\Sigma}_1^1[/math]-determinancy holds, since a measureable cardinal implies the existence of sharps and [math]\displaystyle\mathbf{\Sigma}_1^1=\bigcup_{x\in \omega^{\omega}}\Sigma_1^1(x)[/math]. This result is sharp since Borel determinacy holds in [math]\mathtt{ZFC}[/math].