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- Title
Full and hat inductive definitions are equivalent in NBG.
- Authors
Sato, Kentaro
- Abstract
A new research project has, quite recently, been launched to clarify how different, from systems in second order number theory extending ACA, 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 ID 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).
- Subjects
NUMBER theory; SET theory; MATHEMATICAL expansion; MATHEMATICAL proofs; FIXED point theory
- Publication
Archive for Mathematical Logic, 2015, Vol 54, Issue 1/2, p75
- ISSN
0933-5846
- Publication type
Article
- DOI
10.1007/s00153-014-0403-x