Here are the answers to yesterday’s puzzle. The first correct solution came from our commenter Leo (comment #18 on yesterday’s post).
The assumptions of the problem were: Everything I say out loud can be deduced from my axioms. My axioms include the ordinary axioms for arithmetic, among other things. And I recently said out loud that “I cannot prove that God does not exist”.
The questions were: Can I prove there is no God? Can I prove there is a God? And is there enough information her to determine whether there actually is a God?
The answers are yes, yes and no: Yes, I can prove there is no God. Yes, I can also prove there is a God. And no, you can’t use any of this to determine whether there is a God.
To explain, I’ll use the phrase “logical system” to refer to a system of axioms sufficiently strong to talk about basic arithmetic (and perhaps a whole lot of other things), together with the usual logical rules of inference. It’s given in the problem statement that I am a logical system.
Here are two background facts about logical systems:
A. An inconsistent logical system can prove anything at all. That’s because it’s tautological that if P is self-contradictory, then any statement of the form “P implies Q” is valid. If I’m inconsistent, that means I can prove at least one statement (call it P) that’s self-contradictory. Then if I want to prove, say, that the moon is made of green cheese, I note that:
- I can prove P
- It’s tautological that “P implies the moon is made of green cheese”
- Therefore I can conclude by modus ponens that the moon is made of green cheese.
B. No consistent logical system can prove its own consistency. This is Godel’s celebrated Second Incompleteness Theorem.
Now here’s the argument: