Can all mathematical facts be theoretically proven?

A reply to your edited post:

But my broader point was that, regardless of how you view truth, it's irresponsible to keep hammering this one interpretation of Godel's incompleteness theorem without also mentioning the completeness theorem. I think if you want to teach laymen about Godel's theorems, you should really phrase incompleteness as telling you that lots and lots of models exist.

I think this can be very helpful in a logic class when giving people a more detailed understanding of the theorems. When I studied them, for example, even though I had followed every step of the argument, I was still baffled by the fact that a statement of the form "Every n is P" could be unprovable even though every individual instance ("1 is P", "2 is P", "3 is P", "4 is P", etc.) was provable. What really helped clarify this for me was understanding the notion of a nonstandard model, and seeing that the universal statement was unprovable because PA couldn't rule out models with extra "nonstandard" numbers.

Having said that: I don't think it's necessary to bring up the completeness theorem for laymen. If anything, I think it just causes confusion because the two theorems use different notions of "completeness."

/r/math Thread Parent