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Explore the reverse mathematics of analytic measurability in this 48-minute conference talk examining the foundations of mathematics through the lens of measurable analytical sets. Delve into ongoing collaborative research with Juan Aguilera and Keita Yokoyama that addresses Simpson's fundamental question about determining the correct subsystem needed to prove analytic measurability, originally demonstrated by Lusin. Learn how Yu's previous work established that ATR0 was both sufficient and necessary for proving the existence of measures for coded Borel sets, and discover how this research extends those findings to analytical sets. Understand the innovative approach that draws inspiration from Solovay's construction of a ZF model where every set is Lebesgue measurable, utilizing ATR0 and pseudo-hierarchies to construct non-standard models that provide essential transfinite information about analytical sets. Examine how the scheme of induction for analytical formulae proves sufficient for establishing regularity properties, while the scheme of analytical comprehension yields classical measurability results. Gain insights into the complete reversal theorems that demonstrate the necessity of these logical frameworks, contributing to our understanding of the logical strength required for fundamental results in measure theory and descriptive set theory.
