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I am struggling with the concept of vacuously true base cases in weak induction.

Question - I would appreciate if readers provides simple examples of induction which use a vacuously true base case.

My hope is that simple examples will help me focus on understanding why vacuously true base cases are sufficient. The example I'm working with now is more complex and distracts from the core misunderstanding (Tao Analysis I ex 2.2.5).

Penelope
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  • Depends on what you mean by vacuously. Consider the hypothesis that for all $~k \in \Bbb{Z^+}, ~\exists ~c_0,c_1, \cdots, c_k \in \Bbb{Q},~$ such that for all $$n \in \Bbb{Z_{\geq 0}}, ~ \sum_{i=0}^n i^k = \sum_{j=0}^k c_jn^{k+1-j}.$$ This (true) hypothesis, which is a preliminary step in deriving the Bernoulli number sequence without Calculus, is immediately seen to be true for the base case of $~n=0.~$ That is, when $~n=0,~$ the formula is true regardless of the values of $~c_0, c_1, \cdots, c_k.$ – user2661923 Sep 24 '23 at 17:41
  • hi @user2661923 I mean it in the sense that the base case statement $S(n$) is applied to a number $a$ to which it is not applicable. More context here https://math.stackexchange.com/questions/4774398/why-is-a-vacuously-true-base-case-valid-for-simple-induction .. I asked this more specific question to avoid the distraction of more complex induction proofs. – Penelope Sep 24 '23 at 20:54
  • @Penelope If your statement (more precisely, your unary predicate I suppose) is “not applicable” for some numbers then you cannot meaningfully ask whether it is true for all $n$, so you need to go back to the drawing board and either change it to make it applicable for all $n$, or restrict your attention to a domain of numbers for which it is applicable. – M W Sep 24 '23 at 21:34

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