Nonmonotonicity in (the Metamathematics of) Arithmetic

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This paper is an attempt to bring together two separated areas of research: classical mathematics and metamathematics on the one side, non-monotonic reasoning on the other. This is done by simulating nonmonotonic “logic” through antitonic theory extensions. In the first half, the specific extension procedure proposed here is motivated informally, partly in comparison with some well-known non-monotonic formalisms. Operators V and, more generally, 𝒰δ are obtained which have some plausibility when viewed as giving nonmonotonic theory extensions. In the second half, these operators are treated from a mathematical and metamathematical point of view. Here an important role is played by 𝒰δ-closed theories and 𝒰δ-fixed points. The last section contains results on V-closed theories which are specific for V.

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