Henkin sentences and local reflection principles for Rosser provability☆

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Abstract

In this paper, we investigate Rosser-type Henkin sentences, namely, sentences asserting their own provability in the sense of Rosser, and local reflection principles based on Rosser provability predicates. First, we give a necessary and sufficient condition that a sentence is a Rosser-type Henkin sentence of some Rosser provability predicate, and prove that any negated Rosser sentence can be a Rosser-type Henkin sentence. Secondly, we prove the existence of a Rosser provability predicate whose Rosser-type Henkin sentences are all provable or refutable. Thirdly, we solve the question raised by Shavrukov, and give a Rosser provability predicate whose local reflection principle is strictly weaker than the usual one. At last, we investigate the hierarchy of partial local reflection principles based on Rosser provability predicates.

MSC

03F03
03F30
03F40
03F45

Keywords

Henkin sentences
Local reflection principles
Rosser provability predicates
Löb's theorem
Rosser sentences

Cited by (0)

☆

This work was partly supported by JSPS KAKENHI Grant Numbers 12J00654 and 26887045. The author would like to thank Makoto Kikuchi and Hidenori Kurokawa for their helpful comments. The author would also like to thank the anonymous referees for their valuable comments and suggestions that greatly contributed to improving the earlier versions of this paper. Indeed, Proposition 3.9, Proposition 5.5, Proposition 6.19 are pointed out by one of the referees.

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