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Provident sets and rudimentary set forcing

A. R. D. Mathias (2015)

Fundamenta Mathematicae

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 ω . 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...

P-sets and minimal right ideals in ℕ*

W. R. Brian (2015)

Fundamenta Mathematicae

Recall that a P-set is a closed set X such that the intersection of countably many neighborhoods of X is again a neighborhood of X. We show that if 𝔱 = 𝔠 then there is a minimal right ideal of (βℕ,+) that is also a P-set. We also show that the existence of such P-sets implies the existence of P-points; in particular, it is consistent with ZFC that no minimal right ideal is a P-set. As an application of these results, we prove that it is both consistent with and independent of ZFC that the shift...

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