Abstract.
We investigate the problem how to lift the non - \(\forall \Sigma^b_1(\alpha)\) - conservativity of \(T^2_2(\alpha)\) over \(S^2_2(\alpha)\) to the expected non - \(\forall \Sigma^b_i(\alpha)\) - conservativity of \(T^{i+1}_2(\alpha)\) over \(S^{i+1}_2(\alpha)\), for \(i > 1\). We give a non-trivial refinement of the “lifting method” developed in [4,8], and we prove a sufficient condition on a \(\forall \Sigma^b_1(f)\)-consequence of \(T_2(f)\) to yield the non-conservation result. Further we prove that Ramsey's theorem, a \(\forall \Sigma^b_1(\alpha)\) - formula, is not provable in \(T^1_2(\alpha)\), and that \(\forall \Sigma^b_j(\alpha)\) - conservativity of \(T^{i+1}_2(\alpha)\) over \(T^{i}_2(\alpha)\) implies \(\forall \Sigma^b_j(\alpha)\) - conservativity of the whole \(T_2(\alpha)\) over \(T^{i}_2(\alpha)\), for any \(j \geq 2\).
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Received: 3 April 1997
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Chiari, M., Krajíček, J. Lifting independence results in bounded arithmetic. Arch Math Logic 38, 123–138 (1999). https://doi.org/10.1007/s001530050118
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DOI: https://doi.org/10.1007/s001530050118