Is Peano Arithmetic the most-commonly used arithmetic system

Question

I am learning mathematical logic. I want to ask whether the current arithmetic system that we are talking about is Peano Arithmetic. For instance, when we say we want to prove some conjectures, such as Goldbach's conjecture, are we saying that we want to prove them in Peano Arithmetic? Thank you so much for your reply.

Answer 1

To be honest, I think that when people do arithmetic on a higher level (such as number theory) they are, most of the time, not all too conscious about exactly what arithmetic they use. Just as someone doing topology isn't too concerned with the intricacies of ZFC set theory, or whether they are using some other set theory.

It's about focus. You just can't spend too much energy thinking about the details of the construction of a building's foundation when you decorate a 4th floor apartment. It just isn't worth it in general.