What do you claim when you say you have a proof?

disciple: Your proof was quite beautiful. speaker: Thank you! disciple: Actually, sometimes, I had a bit a problem following. There was quite a jump from the Fundamental Lemma to Theorem A. speaker: Well, yes, because the time was short. But I can give the details. disciple: Rigorous? speaker: Yes, of course. disciple: And Theorem B? speaker: The Fundamental Lemma also implies Theorem B; you can adapt the proof for Theorem A. disciple: So, you have proven the theorems? speaker: Yes, I have a proof – otherwise I wouldn’t say I have theorems. disciple: Of course. skeptic: I’ve overheared your conversation. I would like to know what you mathematicians mean when you say that you “have a proof”. That does not seem so obvious to me. disciple: That we can demonstrate it rigorously. skeptic: This means in the basis of ZFC? speaker: Yes, you can say that “by default option” this means on the basis of ZFC. skeptic: So you say that you have proof of your theoerems on the basis of ZFC? speaker: Yes ...