# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## The strength of the projective Martin conjecture

### Tom 207 / 2010

Fundamenta Mathematicae 207 (2010), 21-27 MSC: 03D28, 03E35, 28A20. DOI: 10.4064/fm207-1-2

#### Streszczenie

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.

#### Autorzy

• C. T. ChongDepartment of Mathematics
Faculty of Science
National University of Singapore
Singapore 117543
e-mail
• Wei WangDepartment of Philosophy
Sun Yat-sen University
Guangzhou 510275, P.R. China
e-mail
• Liang YuInstitute of Mathematical Sciences
Nanjing University
Nanjing, Jiangsu Province 210093, P.R. China
e-mail

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