arXiv subjects

Elena Nogina

Publications and source records attributed to Elena Nogina.

2 recordsLinked to original sources

arXiv2014

On a Hierarchy of Reflection Principles in Peano Arithmetic

We study reflection principles of Peano Arithmetic PA which are based on both proof and provability. Any such reflection principle in PA is equivalent to either $\Box P\!\rightarrow\! P$ ($\Box P$ stands for `$P$ is provable') or $\Box^k u\!\!:\!\!P\!\rightarrow\! P$ for some $k\geq 0$ ($t:P$ states `$t$ is a proof of $P$'). Reflection principles constitute a non-collapsing hierarchy with respect to their deductive strength $$u\!\!:\!\!P\!\rightarrow\! P\ \ \prec\ \ \Box u\!\!:\!\!P\!\rightarrow\! P\ \ \prec\ \ \Box^2 u\!\!:\!\!P\!\rightarrow\! P \ \ \prec\ \ldots\ \prec\ \ \Box P\!\rightarrow\! P.$$

math.LO↗

arXiv2014

On Logic of Formal Provability and Explicit Proofs

In 1933, Gödel considered two modal approaches to describing provability. One captured formal provability and resulted in the logic GL and Solovay's Completeness Theorem. The other was based on the modal logic S4 and led to Artemov's Logic of Proofs LP. In this paper, we study introduced by the author logic GLA, which is a fusion of GL and LP in the union of their languages. GLA is supplied with a Kripke-style semantics and the corresponding completeness theorem. Soundness and completeness of GLA with respect to the arithmetical provability semantics is established.

math.LO↗