They most certainly do, though! They do tell you something about the consistency of ZFC; they tell you that if the system you're working in is consistent, then

They most certainly do, though! They do tell you something about the consistency of ZFC; they tell you that if the system you're working in is consistent, then so is ZFC. Is that not worth knowing? It does sound a bit like you want something out of formal systems that they just can't give you, which is "absolute" truth. edited to add: It's also worth noting that very, very few mathematicians care about actually formalising proofs in a formal system - it's a niche area. The vast majority of mathematicians go on about their business without giving much thought to ZFC and its axioms at all. Many can't even name them all. (I know this, because they are quite surprised when you tell them what some of the axioms are. Especially the Axiom of Infinity.) Lol, I probably can't either,I suspect I would miss a few if I did it off the top of my head.