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- Large cardinal axioms in #SetTheory are often equivalent to asserting that there exists *well founded models* with various properties, which gives you access to more induction.
- In Predicative Arithmetic by Edward Nelson (https://web.math.princeton.edu/~nelson/books.html) it's shown that by severely weakening induction and then considering what numbers still satisfy inductive properties, you can get a system that might be acceptable in #ultrafinitism.

#MathInduction