This post is a favourite of mine, so I’ll try to keep posting it every year it when Christmas is getting closer.
Just a few months before Christmas! But be prepared when your children start asking you whether Santa really exists or not. It’s not as easy to convince them as it once were. The solution to convincing today’s enlightened children is of course to be very rigorous. We need to prove to them that Santa really exists.
So, let’s be pretty formal, and assume that S is the sentence “If S is true, then Santa exists”. That’s just a definition; nothing unusual going on. Seems that if we prove that S is true, then we’ll be done. But we’ll see. Now, the actual logical proof starts.
Suppose S is true. This is just an assumption.
By the definition of S, we can just replace S by its definition, and we get
“If S is true, then Santa exists” is true.
Well, not much gained yet. Probably we’re just warming up. But we can in fact use the assumption, “S is true” once more, together with that. Then we get “Santa exists”. Not bad! But this is of course only because we assumed that S is true. So we’re not there yet. Let’s summarize what we got from the assumption:
“If S is true, then Santa exists”. OK, well, this is the same as what S itself says. Finally something; we’ve proved S itself to be true!
But wait, if S is true, and “If S is true, then Santa exists” is also true, then obviously Santa exists. Done!
So, just sit down together, the whole family, a few days before Christmas, and carefully go through this proof, and you have removed one uncertainty from the celebrations. Also you need to know that there are also grownups who haven’t understood this fact yet.
This is my contribution for the people out there who still want to celebrate that old-fashioned Christmas!
(The proof freely from Boolos and Jeffrey, “Computability and Logic”.)