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# Mathematical Research Letters

## Volume 22 (2015)

### Number 1

### Scales on $\Pi^2_1$ sets

Pages: 301 – 316

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n1.a15

#### Author

#### Abstract

Assuming $\mathsf{AD}^{+} + \theta_0 \lt \Theta \;$ we construct scales of optimal complexity on $\Pi^2_1$ sets of reals. Namely, the norms of the scale are all ordinal-definable (although the scale itself may not be). This paper extends work of Martin and Woodin from the 1980s as well as more recent work of Jackson. The results of this paper were proved in the author’s thesis for more general pointclasses and are presented here for the representative case of the pointclass $\Pi^2_1$.

#### Keywords

determinacy, Suslin set, ordinal-definable, scale

#### 2010 Mathematics Subject Classification

Primary 03E60. Secondary 03E15.