“If this sentence is true, then Santa Claus exists.”

As before, imagine that the antecedent is true – in this case, “this sentence is true”. Does Santa Claus exist, in that case? Well, if the sentence is true, then what it says is true: namely that if the sentence is true, then Santa Claus exists. Therefore, without necessarily believing that Santa Claus exists, or that the sentence is true, it seems we should agree that if the sentence is true, then Santa Claus exists.

But then this means the sentence is true. So Santa Claus does exist. Furthermore we could substitute any claim at all for “Santa Claus exists”. This is Curry’s paradox.

Mark: there are several species of snake called adder.

Maybe this is why Mark called his little story a “joke”.

Just a pet peeve, but maybe you should have said “The idea is that if you can prove that {IF something is true for some ARBITRARY integer n, THEN it is also true for n+1} and {the thing is true for some specific integer k}, then it has to be true for all integers greater than k.

Maybe the math majors can back me up on this. (sorry for nit-picking)

I agree with your nit pick. Saying “if you can prove that something is true for some integer n, and that it is also true for n+1, then it has to be true for all integers greater than n”, as Julianne did, makes it sound like anything that is true for, say, 1 and 2, must also be true for all greater integers. In fact the key point is that being true for any n implies it’s true for n+1. This and the fact that it’s true for some specific n are what you need for induction.

Of course, Julianne surely knows this and was just being sloppy with language.

Ramsey’s theorem. It is accessible to highs school students. A special case of Ramsey’s theorem is the Party Problem : If you invite too many people for a party, you’ll either have more than n people who all know each other or more than m people who do not know each other. For given n and m there exists a number
R(n,m), the so-called Ramsey’s number, which is the minimum number of people you must invite for the party for this to be true.

Surprisingly, even for small values of n and m, not much is known about the Ramsey numbers. E.g. the value of R(5,5) is not known.

One elephant began to play
Upon a spider’s web one day,
He found it such tremendous fun
That he called on another elephant to come.

Two elephants began to play…
(etc.)

It works even better in Spanish: the Spanish version of this song changes the third line, the reason for calling the n+1 elephant, to “Since they saw it resisted…”. So the song allows us to conclude explicitly by induction that a spider web can resist the weight of an infinite number of elephants!