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Self-reference in arithmetic. I. (English) Zbl 1337.03008
Summary: A Gödel sentence is often described as a sentence saying about itself that it is not provable, and a Henkin sentence as a sentence stating its own provability. We discuss what it could mean for a sentence of arithmetic to ascribe to itself a property such as provability or unprovability. The starting point will be the answer Kreisel gave to Henkin’s problem. We describe how the properties of the supposedly self-referential sentences depend on the chosen coding, the formulae expressing the properties and the way a fixed points for the formulae are obtained. This paper is the first of two papers. In the present paper we focus on provability. In Part II [the authors, ibid. 7, No. 4, 671–691 (2014; Zbl 1337.03008)], we will consider other properties like Rosser provability and partial truth predicates.

MSC:
03A05 Philosophical and critical aspects of logic and foundations
03F40 Gödel numberings and issues of incompleteness
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