The answer to Friday’s puzzle is YES. If I am a logic machine who only states what I can prove, and if I say “If I can prove there is no God, then there is no God”, it does follow that I can prove there is no God.
Once again, it was our commenter Leo who got this first (in Friday’s comment thread, graciously rot-13’d).
As with Thursday’s solution to Wednesday’s puzzle, there are two key relevant background facts:
A) An inconsistent system can prove anything.
B) A sufficiently complex consistent system cannot prove its own consistency. (This is Godel’s second incompleteness theorem.)
Here’s the logic:
1) I’ve asserted that “if I can prove there is no God, then there is no God”. We know that I assert only things that I can prove. Therefore I can prove this assertion.
2) That means I can also prove the equivalent assertion that “if there is a God, I cannot prove otherwise”.
3) Therefore, if I take my axioms and add the axiom “There is a God”, then I can prove that there is something I cannot prove.
4) Therefore, if I take my axioms and add the axiom “There is a God”, then I can prove that my axiom system is consistent (by Background Fact A.)
5) Therefore, if I take my axioms and add the axiom “There is a God”, my axiom system is inconsistent. (Because only an inconsistent system can prove its own consistency — that is, Background Fact B.)
6) Therefore the statement “There is a God” must contradict my axiom system.
7) This can happen only if my axiom system is able to prove that “There is no God”.
So yes, I can prove there is no God.