A note on fragments of uniform reflection in second order arithmetic

2022/07/24 by Frittaion, Emanuele
#FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2207.11693

Abstract

We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories of second order arithmetic. The main result is that for any second order arithmetic theory T₀ extending \sf RCA₀ and axiomatizable by a Π¹_k+2 sentence, and for any n≥ k+1, T₀+ RFN_\varPi¹_n+2(T) = T₀ + TI_\varPi¹n(ε₀), T₀+ RFN_\varSigma¹_n+1(T) = T₀+ TI_\varPi¹n(ε₀)^-, where T is T₀ augmented with full induction, and TI_\varPi¹n(ε₀)^- denotes the schema of transfinite induction up to ε₀ for \varPi¹_n formulas without set parameters.

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