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Published online by Cambridge University Press: 12 March 2014
Call a family 
of subsets of a set E inductive if 
and is closed under unions with disjoint singletons, that is, if
A Frege structure is a pair (E, ν) with ν a map to E whose domain dom(ν) is an inductive family of subsets of E such that
In [2] it is shown in a constructive setting that each Frege structure determines a subset which is the domain of a model of Peano's axioms. In this note we establish, within the same constructive setting, three facts. First, we show that the least inductive family of subsets of a set E is precisely the family of decidable Kuratowski finite subsets of E. Secondly, we establish that the procedure presented in [2] can be reversed, that is, any set containing the domain of a model of Peano's axioms determines a map which turns the set into a minimal Frege structure: here by a minimal Frege structure is meant one in which dom(ν) is the least inductive family of subsets of E. And finally, we show that the procedures leading from minimal Frege structures to models of Peano's axioms and vice-versa are mutually inverse. It follows that the postulation of a (minimal) Frege structure is constructively equivalent to the postulation of a model of Peano's axioms.
All arguments will be formulated within constructive (intuitionistic) set theory.
