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The search for new axioms in the hyperuniverse programme. (English) Zbl 1436.03261

Boccuni, Francesca (ed.) et al., Objectivity, realism, and proof. FilMat studies in the philosophy of mathematics. Cham: Springer. Boston Stud. Philos. Hist. Sci. 318, 165-188 (2016).
Summary: The Hyperuniverse Programme, introduced in [T. Arrigoni and S.-D. Friedman, Bull. Symb. Log. 19, No. 1, 77–96 (2013; Zbl 1307.03003)], fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the ‘maximal iterative concept’, and the programme identifies higher-order statements motivated by the maximal iterative concept. The satisfaction of these statements \((\mathbb{H}\)-axioms) in countable transitive models, the collection of which constitutes the ‘hyperuniverse’ \((\mathbb{H})\), has remarkable first-order consequences, some of which we review in Sect. 10.5.
For the entire collection see [Zbl 1351.00011].

MSC:

03E35 Consistency and independence results
03A05 Philosophical and critical aspects of logic and foundations
03E30 Axiomatics of classical set theory and its fragments
03E65 Other set-theoretic hypotheses and axioms

Citations:

Zbl 1307.03003
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References:

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