Homogeneously Suslin set

In descriptive set theory, a set [math]\displaystyle{ S }[/math] is said to be homogeneously Suslin if it is the projection of a homogeneous tree. [math]\displaystyle{ S }[/math] is said to be [math]\displaystyle{ \kappa }[/math]-homogeneously Suslin if it is the projection of a [math]\displaystyle{ \kappa }[/math]-homogeneous tree. If [math]\displaystyle{ A\subseteq{}^\omega\omega }[/math] is a [math]\displaystyle{ \mathbf{\Pi}_1^1 }[/math] set and [math]\displaystyle{ \kappa }[/math] is a measurable cardinal, then [math]\displaystyle{ A }[/math] is [math]\displaystyle{ \kappa }[/math]-homogeneously Suslin. This result is important in the proof that the existence of a measurable cardinal implies that [math]\displaystyle{ \mathbf{\Pi}_1^1 }[/math] sets are determined.

See also

• Projective determinacy

References

• Martin, Donald A. and John R. Steel (Jan 1989). "A Proof of Projective Determinacy". Journal of the American Mathematical Society (American Mathematical Society) 2 (1): 71–125. doi:10.2307/1990913.