$omega_{1}$ under $Pi_{1}$ -Collection

preprint2015arXivOpen access

We describe a proof-theoretic bound on $Sigma_{2}$ -definable countable ordinals in Kripke-Platek set theory with $Pi_{1}$ -Collection and the existence of $omega_{1}$ .

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$omega_{1}$ under $Pi_{1}$ -Collection

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omega1omega_{1}omega1​ under Pi1Pi_{1}Pi1​-Collection

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$omega_{1}$ under $Pi_{1}$ -Collection