If you trust a theory T about integers, and T proves that α is well-ordered, then you should trust T+α, where T+α := T ∪ {Sound(T+β} | β<α}.
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Replying to @ModelOfTheory
A theory T(α) such that forall N, Sound(T+N) iff Busy_Beaver(N) < α.
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Replying to @TheMichaelBurge
Not possible for finite α, because if T+N is sound, then so is T+N+1.
3:12 PM - 24 Apr 2018
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