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Theories of arithmetic in first-order logic, such as Peano arithmetic, second-order arithmetic, Heyting arithmetic, and their subsystems and extensions. Models of Peano arithmetic.

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Which ordinals can be proof-theoretic ordinals of a reasonable theory?

$\newcommand{\bomega}{\boldsymbol\omega}$Given the definition of bounding ordinal in the post and the potential sensitivity to coding mentioned in edit 2, these seem to be two main ways to formalize b …
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