Q&A: Logic and Gödel
Logic and Gödel
Question
Good evening!
The Rabbi always repeats that the problem with logical contradictions is that the human mind cannot hold such a contradiction, and it is nonsense.
But seemingly, according to Gödel’s incompleteness theorems, doesn’t it come out that we are forced to hold a logically hidden worldview? That is, according to Gödel, it is impossible to prove the axioms themselves from within themselves, so if so, doesn’t it follow that our axioms generate a contradiction?
Answer
The incompleteness theorems do not lead us to hold a contradiction; otherwise those theorems would be proof that there is no mathematics. They speak about provability, not truth.
“It is impossible to prove the axioms from within themselves”—I do not even understand what that means, and it certainly is not connected to Gödel’s theorems.
The axioms do not generate any contradiction, because otherwise there would be no mathematics.
Discussion on Answer
One can hold a coherent view, that is, one free of contradictions. True, it is probably not complete. An incomplete world or an incomplete view is not a contradiction.
But they say that one cannot hold a complete coherent worldview; doesn’t it follow from that that we perceive the world as incomplete, and isn’t that itself a contradiction?