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I have a hard time understanding/justifying the proper usage of "vacuously true" starting points in set theory reasoning.

E.g.,

For transfinite induction, one does not have the requirement of a starting point.

If p(a) true for all a<b implies P(b), then P(x) true for all ordinals.

In regular induction, one has both P(0) and P(n)=>P(n+1).

In transfinite induction the starting point P(0) is handwaved as being vacuously true. But then, it is vacuously false, too.

The condition for transfinite induction is obviously much stronger than that for regular induction, and possibly no P can be such that (P(0) - false) and (induction condition -true). That, I would understand. But how can a vacuously true statement - which is equally a vacuously false statement be the starting point for any true/correct judgement?

Can anyone shed some light on this issue?

E.g.,

For transfinite induction, one does not have the requirement of a starting point.

If p(a) true for all a<b implies P(b), then P(x) true for all ordinals.

In regular induction, one has both P(0) and P(n)=>P(n+1).

In transfinite induction the starting point P(0) is handwaved as being vacuously true. But then, it is vacuously false, too.

The condition for transfinite induction is obviously much stronger than that for regular induction, and possibly no P can be such that (P(0) - false) and (induction condition -true). That, I would understand. But how can a vacuously true statement - which is equally a vacuously false statement be the starting point for any true/correct judgement?

Can anyone shed some light on this issue?

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