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BD-N is a weak principle of constructive analysis. Several interesting principles implied by BD-N have already been identified, namely the closure of the anti-Specker spaces under product, the Riemann Permutation Theorem, and the Cauchyness of all partially Cauchy sequences. Here these are shown to be strictly weaker than BD-N, yet not provable in set… (More)
The regular extension axiom, REA, was first considered by Peter Aczel in the context of Constructive Zermelo-Fraenkel Set Theory as an axiom that ensures the existence of many inductively defined sets. REA has several natural variants. In this note we gather together metamathematical results about these variants from the point of view of both classical and… (More)
Constructive ZF + full Separation is shown to be equiconsistent with Second Order Arithmetic.
In order to build the collection of Cauchy reals as a set in constructive set theory, the only power set-like principle needed is exponentiation. In contrast, the proof that theDedekind reals form a set has seemed to requiremore than that. Themain purpose here is to show that exponentiation alone does not suffice for the latter, by furnishing a Kripke model… (More)
When making the transition from a classical to an intuitionistic system, one is compelled to reconsider the axioms chosen. For instance, it would be natural to re-formulate those that imply Excluded Middle. In the context of set theory, typically the Axiom of Foundation is substituted by the Axiom of Set Induction, the two being equivalent classically but… (More)
Some models of set theory are given which contain sets that have some of the important characteristics of being geometric, or spatial, yet do not have any points, in various ways. What’s geometrical is that there are functions to these spaces defined on the ambient spaces which act much like distance functions, and they carry normable Riesz spaces which act… (More)
Intuitionistic set theory without choice axioms does not prove that every Cauchy sequence of rationals has a modulus of convergence, or that the set of Cauchy sequences of rationals is Cauchy complete. Several other related non-provability results are also shown.
We give the natural topological model for ¬BD-N, and use it to show that the closure of spaces with the anti-Specker property under product does not imply BD-N. Also, the natural topological model for ¬BD is presented. Finally, for some of the realizability models known indirectly to falsify BD-N, it is brought out in detail how BD-N fails. keywords:… (More)