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  1. Notions of Invariance for Abstraction Principles.G. A. Antonelli - 2010 - Philosophia Mathematica 18 (3):276-292.
    The logical status of abstraction principles, and especially Hume’s Principle, has been long debated, but the best currently availeble tool for explicating a notion’s logical character—permutation invariance—has not received a lot of attention in this debate. This paper aims to fill this gap. After characterizing abstraction principles as particular mappings from the subsets of a domain into that domain and exploring some of their properties, the paper introduces several distinct notions of permutation invariance for such principles, assessing the philosophical significance (...)
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  • Frege Meets Zermelo: A Perspective on Ineffability and Reflection: A Perspective on Ineffability and Reflection.Stewart Shapiro - 2008 - Review of Symbolic Logic 1 (2):241-266.
    _1. Philosophical background: iteration, ineffability, reflection._ There are at least two heuristic motivations for the axioms of standard set theory, by which we mean, as usual, first-order Zermelo–Fraenkel set theory with the axiom of choice : the iterative conception and limitation of size. Each strand provides a rather hospitable environment for the hypothesis that the set-theoretic universe is ineffable, which is our target in this paper, although the motivation is different in each case.
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