Rudimentary recursion and provident sets
نویسندگان
چکیده
This paper, a contribution to “micro set theory”, is the study promised by the first author in [M4], but improved and extended by work of the second. We use the rudimentarily recursive (set theoretic) functions and the slightly larger collection of gentle functions to develop the theory of provident sets, which are transitive models of PROVI, a very weak subsystem of KP which nevertheless supports not only familiar recursive definitions, such as rank, transitive closure and ordinal addition—though not ordinal multiplication—but also the fundamental definitions of forcing; for a sequel, Provident sets and rudimentary set forcing [MB], will show that forcing can be done over any transitive set or class M which is provident, with a poset that is a member of M; that the Forcing Theorem will hold for ∆̇0 sentences of the forcing language; and that the generic extension will be provident.
منابع مشابه
Provident sets and rudimentary set forcing
Using the theory of rudimentary recursion and provident sets developed in a previous paper, 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 Jω. Much of the development is rudimentary or at worst given by rudimentary...
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