Comparing WO$(\omega^\omega)$ with $\Sigma^0_2$ induction

Stephen G. Simpson

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arXiv · 1508.02655

Comparing WO$(\omega^\omega)$ with $\Sigma^0_2$ induction

Stephen G. Simpson

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Abstract

Let WO$(\omega^\omega)$ be the statement that the ordinal number $\omega^\omega$ is well ordered. WO$(\omega^\omega)$ has occurred several times in the reverse-mathematical literature. The purpose of this expository note is to discuss the place of WO$(\omega^\omega)$ within the standard hierarchy of subsystems of second-order arithmetic. We prove that WO$(\omega^\omega)$ is implied by I$\Sigma^0_2$ and independent of B$\Sigma^0_2$. We also prove that WO$(\omega^\omega)$ and B$\Sigma^0_2$ together do not imply I$\Sigma^0_2$.

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BibTeX RIS

Stephen G. Simpson. 2015-08-11. Comparing WO$(\omega^\omega)$ with $\Sigma^0_2$ induction. https://arxiv.org/abs/1508.02655

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