The following writeup is erroneous. I leave it here purely for the benefit of other fools like me who do not understand induction
(and as of now
, nine people have upvoted this, so there are some). Details at the end.
The mistake that glared at me when I read the original writeup came here:
"...we're left with n people in a room, one of them (in fact, all but one of them!) a redhead. By the hypothesis
, we deduce that all n are redheads, in particular the remaining person."
(i.e. 1 person in room is a redhead, hypothesis states that if 1 person is redheaded all people in the room are, therefore all people in the room are redheaded)
This seems to be using the original "untheorem" to prove itself; you can't do that
. This is a logical fallacy known as "begging the question
". If it was valid, you could make the following, simpler argument:
Hypothesis: If one person in the room is redheaded, all of them are
Presume the existence of a room with n people in it, one of them redheaded
By the hypothesis, if one person is redheaded, all n of them are
Therefore, if one person is redheaded, all of them are, and the hypothesis is true
(actually, this is essentially the argument form used, as far as I can see, and the rest is simple handwaving
has informed me, albeit more politely, that I am in fact talking utter bollocks
, due to my lack of sound mathematical training. When using induction
, I now understand, once you have (un)proved the (un)theorem for the smallest possible value (in this case, one person), you can then take the theorem with regard to n to be true for the purpose of proving the theorem with regard to n+1. Since this was the manner in which the assumption was used, my criticism is wrong.