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We give a simple structural property which characterizes the r.e. sets whose (Turing) degrees are cappable. Since cappable degrees are incomplete, this may be viewed as a solution of Post's program, which asks for a simple structural property of nonrecursive r.e. sets which ensures incompleteness.
It is shown that the propositional modal logic IRM (interpretability logic with Montagna's principle and with witness comparisons in the style of Guaspari's and Solovay's logicR) is sound and complete as the logic ofII
1-conservativity over each∑
1-sound axiomatized theory containingI∑
We define certain properties of subsets of models of arithmetic related to their codability in end extensions and elementary end extensions. We characterize these properties using some more familiar notions concerning cuts in models of arithmetic.
It is shown to be consistent that countable, Fréchet,α
1-spaces are first countable. The result is obtained by using a countable support iteration of proper partial orders of lengthω
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