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Let T be a recursively enumerable theory extending Elementary Arithmetic EA. L. D. Beklemishev proved that the Σ2 local reflection principle for T, (T), is conservative over the Σ1 local reflection principle, (T), with respect to boolean combinations of Σ1-sentences; and asked whether this result is best possible. In this work we answer Beklemishev's question by showing that Π2-sentences are not conserved for T = EA + “f is total,” where f is any nondecreasing computable function with elementary graph. We also discuss how this result generalizes to n > 0 and obtain as an application that for n > 0, is conservative over IΣn with respect to Πn+2-sentences.
Let T be a recursive theory in the language of first order Arithmetic. We prove that if T extends: (a) the scheme of parameter free Δ1-minimization (plus exp). or (b) the scheme of parameter free Π1-induction, then there are no Σ1-maximal models with respect to T. As a consequence, we obtain a new proof of an unpublished theorem of Jeff Paris stating that Σ1-maximal models with respect to IΔ0 + exp do not satisfy the scheme of Σ1-collection BΣ1.
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