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  1. Topological Forcing Semantics with Settling.Robert S. Lubarsky - 2012 - Annals of Pure and Applied Logic 163 (7):820-830.
  • On the Cauchy Completeness of the Constructive Cauchy Reals.Robert S. Lubarsky - 2007 - Mathematical Logic Quarterly 53 (4‐5):396-414.
    It is consistent with constructive set theory (without Countable Choice, clearly) that the Cauchy reals (equivalence classes of Cauchy sequences of rationals) are not Cauchy complete. Related results are also shown, such as that a Cauchy sequence of rationals may not have a modulus of convergence, and that a Cauchy sequence of Cauchy sequences may not converge to a Cauchy sequence, among others.
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  • On the Existence of Stone-Čech Compactification.Giovanni Curi - 2010 - Journal of Symbolic Logic 75 (4):1137-1146.
  • On the Failure of BD-ࡃ and BD, and an Application to the Anti-Specker Property.Robert S. Lubarsky - 2013 - Journal of Symbolic Logic 78 (1):39-56.
    We give the natural topological model for $\neg$BD-${\mathbb N}$, and use it to show that the closure of spaces with the anti-Specker property under product does not imply BD-${\mathbb N}$. Also, the natural topological model for $\neg$BD is presented. Finally, for some of the realizability models known indirectly to falsify BD-$\mathbb{N}$, it is brought out in detail how BD-$\mathbb N$ fails.
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  • A Generalized Cut Characterization of the Fullness Axiom in CZF.Laura Crosilla, Erik Palmgren & Peter Schuster - 2013 - Logic Journal of the IGPL 21 (1):63-76.
    In the present note, we study a generalization of Dedekind cuts in the context of constructive Zermelo–Fraenkel set theory CZF. For this purpose, we single out an equivalent of CZF's axiom of fullness and show that it is sufficient to derive that the Dedekind cuts in this generalized sense form a set. We also discuss the instance of this equivalent of fullness that is tantamount to the assertion that the class of Dedekind cuts in the rational numbers, in the customary (...)
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  • Refinement is Equivalent to Fullness.Albert Ziegler - 2010 - Mathematical Logic Quarterly 56 (6):666-669.
    In the article [4], a new constructive set theoretic principle called Refinement was introduced and analysed. While it seemed to be significantly weaker than its alternative, the more established axiom of Fullness , it was shown to suffice to imply many of the mathematically important consequences. In this article, we will define for each set A a set of truth values which measures the complexity of the equality relation on A. Using these sets we will show that Refinement is actually (...)
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  • Topological Inductive Definitions.Giovanni Curi - 2012 - Annals of Pure and Applied Logic 163 (11):1471-1483.
    In intuitionistic generalized predicative systems as constructive set theory, or constructive type theory, two categories have been proposed to play the role of the category of locales: the category FSp of formal spaces, and its full subcategory FSpi of inductively generated formal spaces. Considered in impredicative systems as the intuitionistic set theory IZF, FSp and FSpi are both equivalent to the category of locales. However, in the mentioned predicative systems, FSp fails to be closed under basic constructions such as that (...)
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  • CZF Does Not Have the Existence Property.Andrew W. Swan - 2014 - Annals of Pure and Applied Logic 165 (5):1115-1147.
    Constructive theories usually have interesting metamathematical properties where explicit witnesses can be extracted from proofs of existential sentences. For relational theories, probably the most natural of these is the existence property, EP, sometimes referred to as the set existence property. This states that whenever ϕϕ is provable, there is a formula χχ such that ϕ∧χϕ∧χ is provable. It has been known since the 80s that EP holds for some intuitionistic set theories and yet fails for IZF. Despite this, it has (...)
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