Abstract
The language of Reflection Calculus \(\mathrm {RC}\) consists of implications between formulas built up from propositional variables and the constant ‘true’ using only conjunction and the diamond modalities which are interpreted in Peano arithmetic as restricted uniform reflection principles. In [6] we introduced \({\mathrm {RC}^\nabla }\), an extension of \(\mathrm {RC}\) by a series of modalities representing the operators associating with a given arithmetical theory T its fragment axiomatized by all theorems of T of arithmetical complexity \(\varPi ^0_n\), for all \(n>0\). In this paper we continue the study of the variable-free fragment of \({\mathrm {RC}^\nabla }\) and characterize its Lindenbaum–Tarski algebra in several natural ways.
L. D. Beklemishev—This work is supported by the Russian Science Foundation under grant 16–11–10252.
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- 1.
We do not distinguish notationally an operation on a set and its restriction to a subset.
- 2.
We avoid the term ‘consistent’, for even the improper theory corresponds to a consistent set of arithmetical sentences.
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Beklemishev, L.D. (2018). A Universal Algebra for the Variable-Free Fragment of \({\mathrm {RC}^\nabla }\) . In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2018. Lecture Notes in Computer Science(), vol 10703. Springer, Cham. https://doi.org/10.1007/978-3-319-72056-2_6
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