The strength of the projective Martin conjecture

Tom 207 / 2010

C. T. Chong, Wei Wang, Liang Yu Fundamenta Mathematicae 207 (2010), 21-27 MSC: 03D28, 03E35, 28A20. DOI: 10.4064/fm207-1-2


We show that Martin's conjecture on $\Pi^1_1$ functions uniformly $\leq_T$-order preserving on a cone implies $\Pi^1_1$ Turing Determinacy over $\hbox{ZF}+{\hbox{DC}}$. In addition, it is also proved that for $n\ge 0$, this conjecture for uniformly degree invariant $\mathbf{\Pi}^1_{2n+1}$ functions is equivalent over ZFC to $\mathbf{\Sigma}^1_{2n+2}$-Axiom of Determinacy. As a corollary, the consistency of the conjecture for uniformly degree invariant $\Pi^1_1$ functions implies the consistency of the existence of a Woodin cardinal.


  • C. T. ChongDepartment of Mathematics
    Faculty of Science
    National University of Singapore
    Lower Kent Ridge Road
    Singapore 117543
  • Wei WangDepartment of Philosophy
    Sun Yat-sen University
    135 Xingang Xi Road
    Guangzhou 510275, P.R. China
  • Liang YuInstitute of Mathematical Sciences
    Nanjing University
    Nanjing, Jiangsu Province 210093, P.R. China

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