Archive for Mathematical Logic

, Volume 54, Issue 1–2, pp 75–112 | Cite as

Full and hat inductive definitions are equivalent in NBG

Article

Abstract

A new research project has, quite recently, been launched to clarify how different, from systems in second order number theory extending ACA0, those in second order set theory extending NBG (as well as those in n + 3-th order number theory extending the so-called Bernays−Gödel expansion of full n + 2-order number theory etc.) are. In this article, we establish the equivalence between \({\Delta^1_0\mbox{\bf-LFP}}\) and \({\Delta^1_0\mbox{\bf-FP}}\), which assert the existence of a least and of a (not necessarily least) fixed point, respectively, for positive elementary operators (or between \({\Delta^{n+2}_0\mbox{\bf-LFP}}\) and \({\Delta^{n+2}_0\mbox{\bf-FP}}\)). Our proof also shows the equivalence between ID1 and \({\widehat{\it ID}_1}\), both of which are defined in the standard way but with the starting theory PA replaced by ZFC (or full n + 2-th order number theory with global well-ordering).

Keywords

Subsystems of Morse–Kelley set theory Von Neumann–Bernays–Gödel set theory Higher order number theory Elementarity of well-foundedness Proof-theoretic strength 

Mathematics Subject Classification

(Primary) 03F35 (Secondary) 03B15 03D65 03E70 03F25 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institut für Inromatik und angewandte MathematikUniversität BernBernSwitzerland

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