Timeline for why is a vacuously true base case valid for simple induction?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Sep 25, 2023 at 20:43 | comment | added | Penelope | I've accepted the answer based on your latest comment which I feel others will benefit from, even if I need to think on it a little more. | |
| Sep 25, 2023 at 20:43 | vote | accept | Penelope | ||
| Sep 25, 2023 at 20:07 | comment | added | M W | @Penelope Sometimes the adjustment to mathematical thinking can be a little head-spinning when starting out. Just keep in mind, when you do come back to this, that ALL “for every” statements are about the nonexistence of counterexamples, not just vacuous ones. $\forall x P(x)$ is identical to $\neg \exists x \neg P(x)$, or in words “$P$ is always true” and “$P$ is never false” have identical meaning in mathematics. | |
| Sep 25, 2023 at 13:55 | comment | added | Penelope | True because "there are no counterexamples" is what I have a problem with. It seems too weak to be the basis of an induction chain. I do thank you for persisting. Perhaps I need to step away and come back to this in a few weeks. My brain is clearly struggling! | |
| Sep 24, 2023 at 23:32 | history | edited | M W | CC BY-SA 4.0 | added 284 characters in body |
| Sep 24, 2023 at 23:18 | comment | added | M W | @Penelope [deleted my last comment because while it was true, it was a little besides the point] I did update the answer with an elaboration on why vacuous truth isn't really anything special. | |
| Sep 24, 2023 at 23:16 | history | edited | M W | CC BY-SA 4.0 | added 1063 characters in body |
| Sep 24, 2023 at 22:40 | comment | added | Penelope | "You may want to expel from your mind the notion that vacuous truth is somehow different from any other kind of truth. Vacuously true statements are just like any other kind." - this is what I'm struggling with. | |
| Sep 24, 2023 at 22:38 | comment | added | Penelope | hi @m-w I've printing out your answer and tried to read it several time. I am still not understanding sadly. Your answer is helpful in showing that an induction proof still has to show the implication is true for the base case (n=1 in your example). However, I can't convince my brain to agree that a vacuously true base case is sufficient. I can see how a non-vacuous base case can start the dominoes falling, but I can't see how a vacuous base case can do this. Perhaps I should come at this from a different angle and ask "what is a vacuously false" statement? | |
| Sep 24, 2023 at 0:03 | history | edited | M W | CC BY-SA 4.0 | deleted 5 characters in body |
| Sep 23, 2023 at 23:29 | history | answered | M W | CC BY-SA 4.0 |