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arXiv · 1405.2558

On a Hierarchy of Reflection Principles in Peano Arithmetic

Elena Nogina
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Abstract

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 $$u\!\!:\!\!P\!\rightarrow\! P\ \ \prec\ \ \Box u\!\!:\!\!P\!\rightarrow\! P\ \ \prec\ \ \Box^2 u\!\!:\!\!P\!\rightarrow\! P \ \ \prec\ \ldots\ \prec\ \ \Box P\!\rightarrow\! P.$$

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Elena Nogina. 2014-05-11. On a Hierarchy of Reflection Principles in Peano Arithmetic. https://arxiv.org/abs/1405.2558

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