A PROOF THAT SANTA CLAUS EXISTS
Consider this sentence:
"If this sentence is true, Santa Claus exists."
Call it P, for short.
Assume, for the sake of argument, that P is true. Then what it says is true: “If P is true, Santa Claus exists.” And we’re assuming P is true. So, Santa Claus exists.
So, we’ve proved that if P is true, Santa Claus exists.
But that’s just what P says!
So, P is true.
So, Santa Claus exists!
This is a version of Curry's paradox. There's obviously a flaw somewhere—can you find it? But there's also obviously something correct about this argument, because Martin Hugo Löb was able to use the same trick to rigorously prove Löb's theorem. To learn what this theorem says, see my blog post:
http://johncarlosbaez.wordpress.com/2013/12/26/logic-probability-and-reflection/
I recently went to a workshop on "logic, probability and reflection" at the Machine Intelligence Research Institute in Berkeley. We studied scientific induction in mathematics and the implications of Löb's theorem to artificial intelligence. My blog post is an introduction to what we did. You can also get links to summary papers on different topics.
#logic
Consider this sentence:
"If this sentence is true, Santa Claus exists."
Call it P, for short.
Assume, for the sake of argument, that P is true. Then what it says is true: “If P is true, Santa Claus exists.” And we’re assuming P is true. So, Santa Claus exists.
So, we’ve proved that if P is true, Santa Claus exists.
But that’s just what P says!
So, P is true.
So, Santa Claus exists!
This is a version of Curry's paradox. There's obviously a flaw somewhere—can you find it? But there's also obviously something correct about this argument, because Martin Hugo Löb was able to use the same trick to rigorously prove Löb's theorem. To learn what this theorem says, see my blog post:
http://johncarlosbaez.wordpress.com/2013/12/26/logic-probability-and-reflection/
I recently went to a workshop on "logic, probability and reflection" at the Machine Intelligence Research Institute in Berkeley. We studied scientific induction in mathematics and the implications of Löb's theorem to artificial intelligence. My blog post is an introduction to what we did. You can also get links to summary papers on different topics.
#logic
