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Provident sets and rudimentary set forcing

Volume 230 / 2015

A. R. D. Mathias Fundamenta Mathematicae 230 (2015), 99-148 MSC: Primary 03E40; Secondary 03E30, 03D65, 03E45. DOI: 10.4064/fm230-2-1

Abstract

Using the theory of rudimentary recursion and provident sets expounded in [MB], we give a treatment of set forcing appropriate for working over models of a theory PROVI which may plausibly claim to be the weakest set theory supporting a smooth theory of set forcing, and of which the minimal model is Jensen's $J_\omega$. Much of the development is rudimentary or at worst given by rudimentary recursions with parameter the notion of forcing under consideration. Our development eschews the power set axiom. We show that the forcing relation for $\dot\varDelta_0 $ wffs is propagated through our hierarchies by a rudimentary function, and we show that the construction of names for the values of rudimentary and rudimentarily recursive functions is similarly propagated. Our main result is that a set-generic extension of a provident set is provident.

Authors

  • A. R. D. MathiasProfesseur émérite, ERMIT, Université de la Réunion
    Address for correspondence:
    Albert-Ludwigs-Universität Freiburg
    Mathematisches Institut
    Abt. f. Mathematische Logik
    Eckerstrasse 1
    D-79104 Freiburg, Germany
    e-mail
    e-mail

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