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Although Wolfram doesn't mention it by name, this is closely related to what he is trying to do: https://en.wikipedia.org/wiki/Reverse_mathematics

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Wanted to thank you for the share, added "Reverse Mathematics: Proofs from the Inside Out" to my reading list which was referenced in the article you posted: https://www.goodreads.com/book/show/34928283-reverse-mathema...

Depending on how comfortable you are with model theory you might also enjoy Dzhafarov and Mummert’s textbook, which first brought the subject to my attention.

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