I took a peek and it is sort of dumb but logically “sound”. Specifically the indictive step.
In the inductive step you assume the statement is true for some number n and use this statement to prove the statement n + 1 is true. If you can do that then you can prove the induction step.
So in this example the statement we assume is true is given n horses, all of them are the same color. To prove the statement for n + 1 horses we look at the n + 1 horses. Then we exclude the last horse. By excluding the last horse we have a set of n horses. By the induction statement this set of horses must all be the same color. So now we’ve proven the first n horses are the same color.
Next we can exclude the first horse. This also gives us a set of n horses. By the induction statement all these horses must also be the same color. Therefore all n + 1 horses must be the same color.
This sounds really dumb but the proof works in the induction step.
The logical issue is that the base case is wrong which is necessary for a complete proof by induction.
I think? I worked through how the induction logic actually fails.
This kind of induction only works if you can actually prove Sets 1 and 2, starting at n and n+1, actually overlap at all stages… and in this case, they don’t.
I took a peek and it is sort of dumb but logically “sound”. Specifically the indictive step.
In the inductive step you assume the statement is true for some number n and use this statement to prove the statement n + 1 is true. If you can do that then you can prove the induction step.
So in this example the statement we assume is true is given n horses, all of them are the same color. To prove the statement for n + 1 horses we look at the n + 1 horses. Then we exclude the last horse. By excluding the last horse we have a set of n horses. By the induction statement this set of horses must all be the same color. So now we’ve proven the first n horses are the same color.
Next we can exclude the first horse. This also gives us a set of n horses. By the induction statement all these horses must also be the same color. Therefore all n + 1 horses must be the same color.
This sounds really dumb but the proof works in the induction step.
The logical issue is that the base case is wrong which is necessary for a complete proof by induction.
I think? I worked through how the induction logic actually fails.
This kind of induction only works if you can actually prove Sets 1 and 2, starting at n and n+1, actually overlap at all stages… and in this case, they don’t.