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AN INCOMPLETENESS THEOREM VIA ORDINAL ANALYSIS
Published online by Cambridge University Press: 12 September 2022
Abstract
We present an analogue of Gödel’s second incompleteness theorem for systems of second-order arithmetic. Whereas Gödel showed that sufficiently strong theories that are $\Pi ^0_1$-sound and
$\Sigma ^0_1$-definable do not prove their own
$\Pi ^0_1$-soundness, we prove that sufficiently strong theories that are
$\Pi ^1_1$-sound and
$\Sigma ^1_1$-definable do not prove their own
$\Pi ^1_1$-soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.
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- © The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
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