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Definability in models of set theory
Published online by Cambridge University Press: 12 March 2014
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Call a set A of ordinals “definable” over a theory T if T is some brand of set theory and whenever A appears in the standard part of a (not necessarily standard) model of T, A is “definable”. Two kinds of “definability” are considered, for each of which is provided a complete (or almost complete) characterization of the hereditarily countable sets of ordinals “definable” over true finitely axiomatizable set theories: (1) there is a single formula ϕ such that in any model of T containing A, A is the unique solution to ϕ; (2) the defining formula is allowed to vary from model to model. (Note. The restrictions “finitely axiomatizable”, and “true” are largely for the sake of convenience: such theories provably have lots of models.)
There are few allusions to what a model theorist would regard as his subject—the methods coming from recursion theory and set theory; but the treatment is intended to be intelligible to nonspecialists. The referee's criticisms have greatly improved the exposition.
I would like to thank Leo Harrington for several discussions, both helpful and hapless, and especially for a clever and timely proof which rescued this project from a moribund state. (Further thanks are due to the Movshon family, as a result of whose New Year's Eve party it became clear that the only really magic formulas are Σ1 formulas.)
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- Copyright © Association for Symbolic Logic 1980
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